{"id":45,"date":"2025-11-08T14:46:03","date_gmt":"2025-11-08T14:46:03","guid":{"rendered":"https:\/\/potentsky.com\/math\/?page_id=45"},"modified":"2026-05-25T09:48:27","modified_gmt":"2026-05-25T09:48:27","slug":"using-the-lambert-w-function","status":"publish","type":"page","link":"https:\/\/potentsky.com\/math\/using-the-lambert-w-function\/","title":{"rendered":"Using the Lambert W Function"},"content":{"rendered":"\n\n\n\n \n \n Solving 5^x = 100x – 175 Using Lambert W Function<\/title>\n <style>\n body {\n font-family: 'Georgia', 'Times New Roman', serif;\n line-height: 1.8;\n max-width: 900px;\n margin: 0 auto;\n padding: 40px 20px;\n background: #f9f9f9;\n color: #333;\n }\n \n .container {\n background: white;\n padding: 40px;\n box-shadow: 0 2px 10px rgba(0,0,0,0.1);\n border-radius: 8px;\n }\n \n h1 {\n color: #1a237e;\n text-align: center;\n font-size: 28px;\n margin-bottom: 30px;\n border-bottom: 3px solid #3f51b5;\n padding-bottom: 15px;\n }\n \n h2 {\n color: #283593;\n font-size: 22px;\n margin-top: 35px;\n margin-bottom: 15px;\n border-left: 5px solid #5c6bc0;\n padding-left: 15px;\n }\n \n h3 {\n color: #3949ab;\n font-size: 18px;\n margin-top: 25px;\n margin-bottom: 12px;\n font-style: italic;\n }\n \n p {\n margin: 15px 0;\n text-align: justify;\n }\n \n .equation-block {\n background: #f5f7fa;\n border-left: 4px solid #3f51b5;\n padding: 20px;\n margin: 20px 0;\n border-radius: 4px;\n overflow-x: auto;\n }\n \n .equation {\n text-align: center;\n font-size: 20px;\n margin: 15px 0;\n font-family: 'Cambria Math', 'Times New Roman', serif;\n }\n \n .inline-math {\n font-family: 'Cambria Math', 'Times New Roman', serif;\n font-style: italic;\n }\n \n .fraction {\n display: inline-flex;\n flex-direction: column;\n vertical-align: middle;\n text-align: center;\n margin: 0 4px;\n }\n \n .fraction .numerator {\n border-bottom: 1.5px solid #333;\n padding: 2px 8px;\n font-size: 0.95em;\n }\n \n .fraction .denominator {\n padding: 2px 8px;\n font-size: 0.95em;\n }\n \n sup {\n font-size: 0.75em;\n vertical-align: super;\n }\n \n sub {\n font-size: 0.75em;\n vertical-align: sub;\n }\n \n .highlight-box {\n background: #e8eaf6;\n border: 2px solid #3f51b5;\n padding: 20px;\n margin: 25px 0;\n border-radius: 6px;\n }\n \n .solution-box {\n background: #fff9c4;\n border: 3px solid #fbc02d;\n padding: 25px;\n margin: 30px 0;\n border-radius: 8px;\n box-shadow: 0 2px 8px rgba(251, 192, 45, 0.3);\n }\n \n .final-answer {\n background: #c8e6c9;\n border: 3px solid #4caf50;\n padding: 30px;\n margin: 35px 0;\n border-radius: 8px;\n text-align: center;\n font-size: 24px;\n font-weight: bold;\n color: #1b5e20;\n }\n \n .check-box {\n background: #e8f5e9;\n border-left: 4px solid #4caf50;\n padding: 15px;\n margin: 15px 0;\n border-radius: 4px;\n }\n \n .step-number {\n display: inline-block;\n background: #3f51b5;\n color: white;\n width: 30px;\n height: 30px;\n line-height: 30px;\n text-align: center;\n border-radius: 50%;\n margin-right: 10px;\n font-weight: bold;\n }\n \n .note {\n background: #fff3e0;\n border-left: 4px solid #ff9800;\n padding: 15px;\n margin: 20px 0;\n border-radius: 4px;\n font-style: italic;\n }\n \n .checkmark {\n color: #4caf50;\n font-weight: bold;\n font-size: 1.2em;\n }\n \n table {\n width: 100%;\n margin: 20px 0;\n border-collapse: collapse;\n }\n \n td {\n padding: 10px;\n text-align: center;\n }\n <\/style>\n<\/head>\n<body>\n <div class=\"container\">\n <h1>Solving 5<sup><i>x<\/i><\/sup> = 100<i>x<\/i> \u2212 175 Using the Lambert W Function<\/h1>\n \n <p class=\"note\">\n <strong>Objective:<\/strong> Solve the equation systematically using the Lambert W method, without guessing solutions.\n <\/p>\n\n <h2><span class=\"step-number\">1<\/span>Rewrite the Equation<\/h2>\n <p>We start with:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 5<sup><i>x<\/i><\/sup> = 100<i>x<\/i> \u2212 175\n <\/div>\n <\/div>\n <p>Using the exponential form 5<sup><i>x<\/i><\/sup> = <i>e<\/i><sup><i>x<\/i> ln 5<\/sup>:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n <i>e<\/i><sup><i>x<\/i> ln 5<\/sup> = 100<i>x<\/i> \u2212 175\n <\/div>\n <\/div>\n\n <h2><span class=\"step-number\">2<\/span>Eliminate the Constant Term in the Linear Part<\/h2>\n <p>We want the right-hand side to be of the form <i>\u03b1<\/i>(<i>x<\/i> \u2212 <i>\u03b2<\/i>). Factor 100:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 100<i>x<\/i> \u2212 175 = 100<span style=\"display: inline-block; margin: 0 5px;\">(<i>x<\/i> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">175<\/span>\n <span class=\"denominator\">100<\/span>\n <\/span>)<\/span> = 100<span style=\"display: inline-block; margin: 0 5px;\">(<i>x<\/i> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n <p>So the equation becomes:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n <i>e<\/i><sup><i>x<\/i> ln 5<\/sup> = 100<span style=\"display: inline-block; margin: 0 5px;\">(<i>x<\/i> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n\n <h2><span class=\"step-number\">3<\/span>Shift the Variable<\/h2>\n <p>Let <span class=\"inline-math\"><i>t<\/i> = <i>x<\/i> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span><\/span>, so <span class=\"inline-math\"><i>x<\/i> = <i>t<\/i> + <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span><\/span>.<\/p>\n \n <p>Substitute into the equation:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n <i>e<\/i><sup>(<i>t<\/i> + <span class=\"fraction\" style=\"font-size: 0.8em;\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>) ln 5<\/sup> = 100<i>t<\/i>\n <\/div>\n <div class=\"equation\">\n <i>e<\/i><sup><i>t<\/i> ln 5<\/sup> \u00b7 <i>e<\/i><sup><span class=\"fraction\" style=\"font-size: 0.8em;\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> ln 5<\/sup> = 100<i>t<\/i>\n <\/div>\n <div class=\"equation\">\n 5<sup><span class=\"fraction\" style=\"font-size: 0.7em;\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span><\/sup> \u00b7 <i>e<\/i><sup><i>t<\/i> ln 5<\/sup> = 100<i>t<\/i>\n <\/div>\n <div class=\"equation\">\n <i>e<\/i><sup><i>t<\/i> ln 5<\/sup> = <span class=\"fraction\">\n <span class=\"numerator\">100<\/span>\n <span class=\"denominator\">5<sup style=\"font-size: 0.7em;\">7\/4<\/sup><\/span>\n <\/span><i>t<\/i>\n <\/div>\n <\/div>\n \n <p>Let <span class=\"inline-math\"><i>k<\/i> = <span class=\"fraction\">\n <span class=\"numerator\">100<\/span>\n <span class=\"denominator\">5<sup style=\"font-size: 0.7em;\">7\/4<\/sup><\/span>\n <\/span><\/span>, so:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\">\n <i>e<\/i><sup><i>t<\/i> ln 5<\/sup> = <i>k t<\/i>\n <\/div>\n <\/div>\n\n <h2><span class=\"step-number\">4<\/span>Rearrange into Lambert W Form<\/h2>\n <p>Divide both sides by <i>k t<\/i> and rearrange:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 1 = <i>k t e<\/i><sup>\u2212<i>t<\/i> ln 5<\/sup>\n <\/div>\n <div class=\"equation\">\n <span class=\"fraction\">\n <span class=\"numerator\">1<\/span>\n <span class=\"denominator\"><i>k<\/i><\/span>\n <\/span> = <i>t e<\/i><sup>\u2212<i>t<\/i> ln 5<\/sup>\n <\/div>\n <\/div>\n \n <p>Multiply both sides by \u2212ln 5:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\"><i>k<\/i><\/span>\n <\/span> = (\u2212<i>t<\/i> ln 5) <i>e<\/i><sup>\u2212<i>t<\/i> ln 5<\/sup>\n <\/div>\n <\/div>\n \n <p>Let <i>u<\/i> = \u2212<i>t<\/i> ln 5, then:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\">\n \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\"><i>k<\/i><\/span>\n <\/span> = <i>u e<sup>u<\/sup><\/i>\n <\/div>\n <\/div>\n\n <h2><span class=\"step-number\">5<\/span>Apply Lambert W Function<\/h2>\n <p>The Lambert W function is defined as the inverse of <i>f<\/i>(<i>w<\/i>) = <i>w e<sup>w<\/sup><\/i>. Therefore:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n <i>u<\/i> = <i>W<\/i><span style=\"display: inline-block; margin: 0 3px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\"><i>k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <div class=\"equation\">\n <i>t<\/i> = \u2212<span class=\"fraction\">\n <span class=\"numerator\">1<\/span>\n <span class=\"denominator\">ln 5<\/span>\n <\/span> \u00b7 <i>W<\/i><span style=\"display: inline-block; margin: 0 3px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\"><i>k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n \n <p>Since <i>x<\/i> = <i>t<\/i> + <sup>7<\/sup>\u2044<sub>4<\/sub>:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\" style=\"font-size: 18px;\">\n <i>x<\/i> = <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">1<\/span>\n <span class=\"denominator\">ln 5<\/span>\n <\/span> \u00b7 <i>W<\/i><span style=\"display: inline-block; margin: 0 3px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\"><i>k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n\n <h2><span class=\"step-number\">6<\/span>Simplify the Constant Inside W<\/h2>\n <p>Recall <i>k<\/i> = <sup>100<\/sup>\u2044<sub>5<sup>7\/4<\/sup><\/sub>. We need to simplify:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\"><i>k<\/i><\/span>\n <\/span> = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5 \u00b7 5<sup style=\"font-size: 0.7em;\">7\/4<\/sup><\/span>\n <span class=\"denominator\">100<\/span>\n <\/span>\n <\/div>\n <\/div>\n \n <p>Note that 5<sup>7\/4<\/sup> = 5<sup>2\u22121\/4<\/sup> = 25 \u00b7 5<sup>\u22121\/4<\/sup>. Therefore:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n <span class=\"fraction\">\n <span class=\"numerator\">ln 5 \u00b7 5<sup style=\"font-size: 0.7em;\">7\/4<\/sup><\/span>\n <span class=\"denominator\">100<\/span>\n <\/span> = <span class=\"fraction\">\n <span class=\"numerator\">ln 5 \u00b7 25 \u00b7 5<sup style=\"font-size: 0.7em;\">\u22121\/4<\/sup><\/span>\n <span class=\"denominator\">100<\/span>\n <\/span> = <span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4 \u00b7 5<sup style=\"font-size: 0.7em;\">1\/4<\/sup><\/span>\n <\/span>\n <\/div>\n <\/div>\n \n <p>Let <span class=\"inline-math\"><i>A<\/i> = <span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4 \u00b7 5<sup style=\"font-size: 0.7em;\">1\/4<\/sup><\/span>\n <\/span><\/span>. Then:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\" style=\"font-size: 18px;\">\n <i>x<\/i> = <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">1<\/span>\n <span class=\"denominator\">ln 5<\/span>\n <\/span> \u00b7 <i>W<\/i>(\u2212<i>A<\/i>)\n <\/div>\n <\/div>\n\n <h2><span class=\"step-number\">7<\/span>Find W(\u2212A) for Both Real Branches<\/h2>\n <p>Numerically:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n <i>A<\/i> \u2248 <span class=\"fraction\">\n <span class=\"numerator\">1.609438<\/span>\n <span class=\"denominator\">4 \u00d7 1.495349<\/span>\n <\/span> \u2248 <span class=\"fraction\">\n <span class=\"numerator\">1.609438<\/span>\n <span class=\"denominator\">5.981396<\/span>\n <\/span> \u2248 0.269\n <\/div>\n <\/div>\n \n <p>So \u2212<i>A<\/i> \u2248 \u22120.269, which is greater than \u2212<sup>1<\/sup>\u2044<sub><i>e<\/i><\/sub> \u2248 \u22120.3679.<\/p>\n \n <div class=\"note\">\n <strong>Important:<\/strong> Since \u2212<i>A<\/i> > \u2212<sup>1<\/sup>\u2044<sub><i>e<\/i><\/sub>, there are <strong>two real branches<\/strong> of the Lambert W function: <i>W<\/i><sub>0<\/sub> (principal branch) and <i>W<\/i><sub>\u22121<\/sub> (secondary branch).\n <\/div>\n\n <h2><span class=\"step-number\">8<\/span>Solve for W(\u2212A) Exactly<\/h2>\n \n <p>We need <i>W<\/i>(\u2212<i>A<\/i>) such that:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n <i>W<\/i>(\u2212<i>A<\/i>) \u00b7 <i>e<\/i><sup><i>W<\/i>(\u2212<i>A<\/i>)<\/sup> = \u2212<i>A<\/i> = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4 \u00b7 5<sup style=\"font-size: 0.7em;\">1\/4<\/sup><\/span>\n <\/span>\n <\/div>\n <\/div>\n\n <h3>First Branch (W<sub>0<\/sub>):<\/h3>\n <p>Try <i>W<\/i><sub>0<\/sub>(\u2212<i>A<\/i>) = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span><\/p>\n \n <p><strong>Check:<\/strong><\/p>\n <div class=\"check-box\">\n <div class=\"equation\">\n <span style=\"display: inline-block; margin: 0 3px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span> \u00b7 exp<span style=\"display: inline-block; margin: 0 3px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span> = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u00b7 5<sup>\u22121\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4 \u00b7 5<sup style=\"font-size: 0.7em;\">1\/4<\/sup><\/span>\n <\/span> = \u2212<i>A<\/i> <span class=\"checkmark\">\u2713<\/span>\n <\/div>\n <\/div>\n \n <p>Therefore: <i>W<\/i><sub>0<\/sub>(\u2212<i>A<\/i>) = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span><\/p>\n \n <div class=\"solution-box\">\n <div class=\"equation\" style=\"font-size: 20px; color: #f57c00;\">\n <i>x<\/i><sub>1<\/sub> = <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">1<\/span>\n <span class=\"denominator\">ln 5<\/span>\n <\/span> \u00b7 <span style=\"display: inline-block; margin: 0 5px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span> = <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> + <span class=\"fraction\">\n <span class=\"numerator\">1<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> = <span style=\"font-size: 1.3em; color: #d32f2f;\">2<\/span>\n <\/div>\n <\/div>\n\n <h3>Second Branch (W<sub>\u22121<\/sub>):<\/h3>\n <p>Try <i>W<\/i><sub>\u22121<\/sub>(\u2212<i>A<\/i>) = \u2212<span class=\"fraction\">\n <span class=\"numerator\">5 ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span><\/p>\n \n <p><strong>Check:<\/strong><\/p>\n <div class=\"check-box\">\n <div class=\"equation\">\n <span style=\"display: inline-block; margin: 0 3px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">5 ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span> \u00b7 exp<span style=\"display: inline-block; margin: 0 3px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">5 ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span> = \u2212<span class=\"fraction\">\n <span class=\"numerator\">5 ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u00b7 5<sup>\u22125\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u00b7 5 \u00b7 5<sup>\u22125\/4<\/sup> = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u00b7 5<sup>\u22121\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212<span class=\"fraction\">\n <span class=\"numerator\">ln 5<\/span>\n <span class=\"denominator\">4 \u00b7 5<sup style=\"font-size: 0.7em;\">1\/4<\/sup><\/span>\n <\/span> = \u2212<i>A<\/i> <span class=\"checkmark\">\u2713<\/span>\n <\/div>\n <\/div>\n \n <p>Therefore: <i>W<\/i><sub>\u22121<\/sub>(\u2212<i>A<\/i>) = \u2212<span class=\"fraction\">\n <span class=\"numerator\">5 ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span><\/p>\n \n <div class=\"solution-box\">\n <div class=\"equation\" style=\"font-size: 20px; color: #f57c00;\">\n <i>x<\/i><sub>2<\/sub> = <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> \u2212 <span class=\"fraction\">\n <span class=\"numerator\">1<\/span>\n <span class=\"denominator\">ln 5<\/span>\n <\/span> \u00b7 <span style=\"display: inline-block; margin: 0 5px;\">(\u2212<span class=\"fraction\">\n <span class=\"numerator\">5 ln 5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span>)<\/span> = <span class=\"fraction\">\n <span class=\"numerator\">7<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> + <span class=\"fraction\">\n <span class=\"numerator\">5<\/span>\n <span class=\"denominator\">4<\/span>\n <\/span> = <span style=\"font-size: 1.3em; color: #d32f2f;\">3<\/span>\n <\/div>\n <\/div>\n\n <h2><span class=\"step-number\">9<\/span>Final Solutions<\/h2>\n <div class=\"final-answer\">\n <i>x<\/i> = 2 and <i>x<\/i> = 3\n <\/div>\n\n <h2>Verification<\/h2>\n <div class=\"highlight-box\">\n <h3>For <i>x<\/i> = 2:<\/h3>\n <table>\n <tr>\n <td><strong>Left side:<\/strong><\/td>\n <td>5<sup>2<\/sup> = 25<\/td>\n <\/tr>\n <tr>\n <td><strong>Right side:<\/strong><\/td>\n <td>100(2) \u2212 175 = 200 \u2212 175 = 25<\/td>\n <\/tr>\n <tr>\n <td colspan=\"2\" style=\"color: #4caf50; font-weight: bold; font-size: 1.2em;\">25 = 25 \u2713<\/td>\n <\/tr>\n <\/table>\n \n <h3>For <i>x<\/i> = 3:<\/h3>\n <table>\n <tr>\n <td><strong>Left side:<\/strong><\/td>\n <td>5<sup>3<\/sup> = 125<\/td>\n <\/tr>\n <tr>\n <td><strong>Right side:<\/strong><\/td>\n <td>100(3) \u2212 175 = 300 \u2212 175 = 125<\/td>\n <\/tr>\n <tr>\n <td colspan=\"2\" style=\"color: #4caf50; font-weight: bold; font-size: 1.2em;\">125 = 125 \u2713<\/td>\n <\/tr>\n <\/table>\n <\/div>\n\n <p style=\"text-align: center; margin-top: 40px; font-style: italic; color: #666;\">\n This solution demonstrates the power of the Lambert W function in solving transcendental equations that combine exponential and linear terms.\n <\/p>\n <\/div>\n<\/body>\n<\/html>\n\n\n\nYou know exactly which branch to use by comparing your simplified output value to \\(-1\\), which is the boundary value at the branch point \\(z = -1\/e\\).Every real value of the Lambert \\(W\\) function falls strictly into one of two zones:The Principal Branch (\\(W_{0}\\)): Always outputs values greater than or equal to \\(-1\\) (Interval: \\([-1, \\infty)\\)).The Lower Branch (\\(W_{-1}\\)): Always outputs values less than or equal to \\(-1\\) (Interval: \\((-\\infty, -1]\\)).By evaluating the exact expressions you found against \\(-1\\), the branch choice becomes definitive.Branch Verification for Your Solutions1. Verifying the First Value: \\(-\\frac{\\ln 5}{4}\\)Approximate value: Since \\(\\ln 5 \\approx 1.609\\), then \\(-\\frac{1.609}{4} \\approx \\mathbf{-0.402}\\).Comparison: \\(-0.402 > -1\\).Conclusion: Because this value is greater than \\(-1\\), it must belong to the principal branch, meaning \\(W_0(-A) = -\\frac{\\ln 5}{4}\\).2. Verifying the Second Value: \\(-\\frac{5}{4}\\ln 5\\)Approximate value: This is \\(-1.25 \\times 1.609 \\approx \\mathbf{-2.012}\\).Comparison: \\(-2.012 < -1\\).Conclusion: Because this value is less than \\(-1\\), it must belong to the lower branch, meaning \\(W_{-1}(-A) = -\\frac{5}{4}\\ln 5\\).\ud83d\udca1 The Algebraic Trick to Spotting CandidatesIf you ever want to \"reverse-engineer\" these exact forms without guessing, look at your original equation rewritten in the form \\(Y e^Y = -A\\).You had:\\(-A=-\\frac{\\ln 5}{4\\cdot 5^{1\/4}}=\\left(-\\frac{\\ln 5}{4}\\right)\\cdot 5^{-1\/4}\\)To match \\(Ye^{Y}\\), you need the base \\(5^{-1\/4}\\) to look like \\(e^{Y}\\):For \\(W_{0}\\): Rewrite \\(5^{-1\/4}\\) as \\(e^{-\\frac{\\ln 5}{4}}\\). This makes \\(Y = -\\frac{\\ln 5}{4}\\).For \\(W_{-1}\\): Multiply the front and the exponent by \\(\\frac{5}{\\ln 5}\\) variants to find the second hidden match, which yields \\(Y = -\\frac{5 \\ln 5}{4}\\).Once you extract those two mathematical candidates, you simply apply the \\(\\ge -1\\) or \\(\\le -1\\) rule to assign them to \\(W_{0}\\) or \\(W_{-1}\\).Would you like to try another equation to practice finding the exact analytical forms of \\(W\\), or see how this applies to complex branches (\\(W_{k}\\))?You said: I don't understand this identity:... Note: W(y)\/y = e^[-W(y)] ...This identity looks confusing at first, but it is actually a direct rearrangement of the definitional property you looked at earlier: \\(W(y)e^{W(y)} = y\\).Here is the quick, 3-step proof showing exactly where it comes from:The 3-Step DerivationStart with the definition:\\(W(y)e^{W(y)}=y\\)Divide both sides by \\(y\\):\\(\\frac{W(y)e^{W(y)}}{y}=1\\)Divide both sides by \\(e^{W(y)}\\) to isolate the fraction:\\(\\frac{W(y)}{y}=\\frac{1}{e^{W(y)}}\\)Using the rule of exponents where \\(\\frac{1}{e^A} = e^{-A}\\), the right side becomes \\(e^{-W(y)}\\).This leaves you with the exact identity:\\(\\frac{W(y)}{y}=e^{-W(y)}\\)Why is this identity useful?In calculus and algebra, this identity is a powerful tool used to move a variable out of a denominator.If you are simplifying an expression and you see \\(\\frac{1}{y}\\) multiplied by \\(W(y)\\), you can instantly swap it out for a clean exponential term \\(e^{-W(y)}\\), which is often much easier to integrate or differentiate.Would you like to see how this specific identity changes the way you take the derivative of the Lambert \\(W\\) function?\n\n\n<div class=\"tonYlb\" data-tr-rsts=\"false\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 24px 0px 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"\" data-streaming-container=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"CKgc1d\" data-scope-id=\"turn\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-subtree=\"aimc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"Zkbeff\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_3\" data-sfc-cb=\"\" data-ved=\"2ahUKEwi_y9X2gNSUAxU-Q6QEHdf2FBUQ2O0OegYIAAgBEAA\" data-hveid=\"CAAIARAA\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"pWvJNd\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"mZJni Dn7Fzd\" dir=\"ltr\" data-container-id=\"main-col\" data-xid=\"VpUvz\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-ved=\"2ahUKEwi_y9X2gNSUAxU-Q6QEHdf2FBUQ3KYQegYIAAgBEAE\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIAhAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">You know exactly which branch to use by comparing your simplified output value to <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\\(-1\\)<!--TgQPHd|[]--><\/strong>, which is the boundary value at the branch point \\(z = -1\/e\\).<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIAxAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Every real value of the Lambert \\(W\\) function falls strictly into one of two zones:<!--TgQPHd|[]--><\/div>\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li style=\"list-style-type: none;\">\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBBAA\" data-sae=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">The Principal Branch (\\(W_{0}\\)):<!--TgQPHd|[]--><\/strong> Always outputs values <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">greater than or equal to \\(-1\\)<!--TgQPHd|[]--><\/strong> (Interval: \\([-1, \\infty)\\)).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBBAE\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">The Lower Branch (\\(W_{-1}\\)):<!--TgQPHd|[]--><\/strong> Always outputs values <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">less than or equal to \\(-1\\)<!--TgQPHd|[]--><\/strong> (Interval: \\((-\\infty, -1]\\)).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!--TgQPHd|[]--><\/p>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBRAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">By evaluating the exact expressions you found against \\(-1\\), the branch choice becomes definitive.<!--TgQPHd|[]--><\/div>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<hr class=\"j3tEEe\" data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 32px 0px; text-decoration: none; border-bottom: 0px rgb(128, 128, 128);\" \/>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div class=\"otQkpb\" role=\"heading\" aria-level=\"3\" data-animation-nesting=\"\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 20px; font-weight: 600; margin: 24px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Branch Verification for Your Solutions<!--TgQPHd|[]--><\/div>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div class=\"AdPoic\" role=\"heading\" aria-level=\"3\" data-animation-nesting=\"\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-sae=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 24px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">1. Verifying the First Value: \\(-\\frac{\\ln 5}{4}\\)<!--TgQPHd|[]--><\/div>\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li style=\"list-style-type: none;\">\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAICRAA\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Approximate value:<!--TgQPHd|[]--><\/strong> Since \\(\\ln 5 \\approx 1.609\\), then \\(-\\frac{1.609}{4} \\approx \\mathbf{-0.402}\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAICRAD\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Comparison:<!--TgQPHd|[]--><\/strong> \\(-0.402 > -1\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAICRAF\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Conclusion:<!--TgQPHd|[]--><\/strong> Because this value is greater than \\(-1\\), it <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">must<!--TgQPHd|[]--><\/strong> belong to the principal branch, meaning \\(W_0(-A) = -\\frac{\\ln 5}{4}\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!--TgQPHd|[]--><\/p>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div class=\"AdPoic\" role=\"heading\" aria-level=\"3\" data-animation-nesting=\"\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 24px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">2. Verifying the Second Value: \\(-\\frac{5}{4}\\ln 5\\)<!--TgQPHd|[]--><\/div>\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li style=\"list-style-type: none;\">\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAICxAA\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Approximate value:<!--TgQPHd|[]--><\/strong> This is \\(-1.25 \\times 1.609 \\approx \\mathbf{-2.012}\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAICxAC\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Comparison:<!--TgQPHd|[]--><\/strong> \\(-2.012 < -1\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAICxAE\" data-sae=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Conclusion:<!--TgQPHd|[]--><\/strong> Because this value is less than \\(-1\\), it <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">must<!--TgQPHd|[]--><\/strong> belong to the lower branch, meaning \\(W_{-1}(-A) = -\\frac{5}{4}\\ln 5\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!--TgQPHd|[]--><\/p>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<hr class=\"j3tEEe\" data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 32px 0px; text-decoration: none; border-bottom: 0px rgb(128, 128, 128);\" \/>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div class=\"otQkpb\" role=\"heading\" aria-level=\"3\" data-animation-nesting=\"\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 20px; font-weight: 600; margin: 24px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\ud83d\udca1 The Algebraic Trick to Spotting Candidates<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIDhAA\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">If you ever want to “reverse-engineer” these exact forms without guessing, look at your original equation rewritten in the form \\(Y e^Y = -A\\).<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIDxAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">You had:<br data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\" \/>\\(-A=-\\frac{\\ln 5}{4\\cdot 5^{1\/4}}=\\left(-\\frac{\\ln 5}{4}\\right)\\cdot 5^{-1\/4}\\)<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIEBAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">To match \\(Ye^{Y}\\), you need the base \\(5^{-1\/4}\\) to look like \\(e^{Y}\\):<!--TgQPHd|[]--><\/div>\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li style=\"list-style-type: none;\">\n<ul class=\"KsbFXc U6u95\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIERAA\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">For \\(W_{0}\\):<!--TgQPHd|[]--><\/strong> Rewrite \\(5^{-1\/4}\\) as \\(e^{-\\frac{\\ln 5}{4}}\\). This makes \\(Y = -\\frac{\\ln 5}{4}\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIERAF\" data-sae=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">For \\(W_{-1}\\):<!--TgQPHd|[]--><\/strong> Multiply the front and the exponent by \\(\\frac{5}{\\ln 5}\\) variants to find the second hidden match, which yields \\(Y = -\\frac{5 \\ln 5}{4}\\).<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p><!--TgQPHd|[]--><\/p>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIEhAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Once you extract those two mathematical candidates, you simply apply the <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\\(\\ge -1\\) or \\(\\le -1\\) rule<!--TgQPHd|[]--><\/strong> to assign them to \\(W_{0}\\) or \\(W_{-1}\\).<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIExAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Would you like to try another equation to practice <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">finding the exact analytical forms<!--TgQPHd|[]--><\/strong> of \\(W\\), or see how this applies to <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">complex branches (\\(W_{k}\\))<!--TgQPHd|[]--><\/strong>?<!--TgQPHd|[]--><\/div>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div data-xid=\"Gd7Hsc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_4n\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"DBd2Wb\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 4px 0px 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[[1,1,1]]--><\/span>\n<div class=\"zkL70c\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"RkJvxe\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_4q\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"YHsVn\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_4s,YmKkte_4t,YmKkte_4u\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: Arial; font-size: 13.3333px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(86, 89, 94);\">\u00a0<\/div>\n<p><!--TgQPHdYmKkte_4p\/eGiyHb|[[3,1],{"u":0},{"s":false,"i":"YmKkte_4p\/eGiyHb"},false,true]--><\/p>\n<\/div>\n<p><!--TgQPHdYmKkte_4m\/wGhr1b|[{"s":null,"i":"YmKkte_4m\/ns9scf"},1,{"s":true,"i":"YmKkte_4m\/wGhr1b"},false]--><\/p>\n<\/div>\n<div data-xid=\"YruvMc\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"OUQe0e\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--qkimaf YmKkte_4y\/HugV6-->\n<div data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div id=\"shrproxy36\" class=\"bQ0Yzc\" data-ved=\"0CAAQp4QQahcKEwjIiK7d_NOUAxUAAAAAHQAAAAAQUg\" data-hveid=\"0\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"\" data-copy-service-computed-style=\"font-family: Arial; font-size: 13.3333px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(86, 89, 94);\">\u00a0<\/div>\n<div data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<\/div>\n<\/div>\n<p><!--cqw1tb YmKkte_4y\/HugV6--><\/p>\n<\/div>\n<p><!--TgQPHdYmKkte_4m\/ns9scf|[[0,null,null,null,0,0,0,null,1],{"s":null,"i":"YmKkte_4m\/ns9scf"},false,34]--><\/p>\n<\/div>\n<div data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_5x,YmKkte_5y,YmKkte_5z\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div id=\"fbproxy37\" class=\"UrecDd\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"YHsVn\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_61,YmKkte_62,YmKkte_63\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><button class=\"cEme7d\" aria-label=\"Good response\" aria-pressed=\"false\" data-snt=\"40\" data-ved=\"2ahUKEwi_y9X2gNSUAxU-Q6QEHdf2FBUQzO0OegYIAAgUEBA\" data-hveid=\"CAAIFBAQ\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: Arial; font-size: 13.3333px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 2px solid rgba(0, 0, 0, 0);\"><\/button><!--TgQPHd|[[1],{"u":0},false,false,true]--><\/div>\n<div class=\"YHsVn\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_65,YmKkte_66,YmKkte_67\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><button class=\"cEme7d\" aria-label=\"Bad response\" aria-pressed=\"false\" data-snt=\"41\" data-ved=\"2ahUKEwi_y9X2gNSUAxU-Q6QEHdf2FBUQze0OegYIAAgUEBE\" data-hveid=\"CAAIFBAR\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: Arial; font-size: 13.3333px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 2px solid rgba(0, 0, 0, 0);\"><\/button><!--TgQPHd|[[1],{"u":0},false,false,true]--><\/div>\n<\/div>\n<p><!--TgQPHd|[[[null,null,null,null,null,null,null,null,null,null,null,null,null,[null,null,null,null,null,null,null,[["aim_labs","false"]]],null,null,null,null,null,114,null,null,null,null,null,null,null,null,null,"RH7zg"],1,null,null,null,0]]--><\/p>\n<\/div>\n<div class=\"YOTKvb\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_69\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-suuid=\"eXTog|6355c75d-e82a-4a7e-953d-ce18514916da|\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_6b\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"DBNuff\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_6d,YmKkte_6e,YmKkte_6f\" data-sfc-cb=\"L8Tm3b=&YmKkte_6a:vSwXaf;ZzAshb=&YmKkte_6a:A9tW8d;qoQ6Mc=&YmKkte_6a:PSl28c\" data-ved=\"2ahUKEwi_y9X2gNSUAxU-Q6QEHdf2FBUQxPMRegYIAAgUEBI\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"vqxIde\" tabindex=\"0\" role=\"button\" aria-label=\"More options\" aria-haspopup=\"listbox\" aria-disabled=\"false\" data-ved=\"2ahUKEwi_y9X2gNSUAxU-Q6QEHdf2FBUQx-0SegYIAAgUEBM\" data-hveid=\"CAAIFBAT\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"YHsVn\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_6h,YmKkte_6i,YmKkte_6j\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"Wu8bab\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 2px solid rgba(0, 0, 0, 0);\">\n<div data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<\/div>\n<p><!--TgQPHdYmKkte_6a\/BXv9hf|[[1],{"u":0},false,{"s":false,"i":"YmKkte_6a\/BXv9hf"},true]--><\/p>\n<\/div>\n<\/div>\n<p><!--TgQPHdYmKkte_4m\/wGhr1b|[[[null,null,null,null,null,null,null,null,null,null,null,null,[308935,null,0]],null,"More options",null,[null,null,null,null,null,null,null,null,null,null,null,null,[308939,null,0]]],{"u":0},false,[-1,-1],{"s":true,"i":"YmKkte_4m\/wGhr1b"}]--><\/p>\n<\/div>\n<p><!--qkimaf YmKkte_6a\/HugV6--><!--cqw1tb YmKkte_6a\/HugV6--><!--qkimaf YmKkte_6a\/WyzG9e--><!--cqw1tb YmKkte_6a\/WyzG9e--><!--TgQPHd|[{"s":true,"i":"YmKkte_4m\/wGhr1b"},{"s":null,"i":"YmKkte_4m\/ns9scf"},1]--><\/p>\n<\/div>\n<p><!--TgQPHd|[false,{"s":null,"i":"YmKkte_4m\/ns9scf"},1,{"s":true,"i":"YmKkte_4m\/wGhr1b"},{"s":null,"i":"YmKkte_4m\/DbG24d"}]--><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-zzy=\"1\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<p><!--TgQPHd|[[1,0,0,null,0,0,1,null,null,null,0,0,0,0,0,1,"Explain how to determine the exact branch (W0 and W-1) to use in the Lambert W function to find the solutions to the equation 5^x \\u003d 100x - 175, given the previous simplification steps and the derived expression x\\u003d7\/4\u2013(1\/ln5)W(\u2212ln5*5^(7\/4)\/100)",0,0,null,0,0,0]]--><\/p>\n<\/div>\n<p><!--TgQPHd|[]--><\/p>\n<\/div>\n<div class=\"SGF5Lb\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"SGF5Lb\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"SGF5Lb\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"SGF5Lb\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"SGF5Lb\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"SGF5Lb\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"SGF5Lb\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<p><!--TgQPHd|[]--><\/p>\n<\/div>\n<div class=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"ofHStc qmNpEc\" data-container-id=\"rhs-col\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-ved=\"2ahUKEwi_y9X2gNSUAxU-Q6QEHdf2FBUQ3aYQegYIAAgBEAI\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[[0,0]]--><\/div>\n<p><!--TgQPHd|[]--><\/p>\n<\/div>\n<div data-xid=\"aQR2uf\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-sfc-root=\"c\" data-wiz-uids=\"YmKkte_b,YmKkte_c\" data-sfc-cb=\"rnAwJc=&YmKkte_1:bREfu\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHdYmKkte_1\/YOIIaf;YmKkte_1\/VWapad|[{"s":false,"i":"YmKkte_1\/YOIIaf"},{"s":null,"i":"YmKkte_1\/VWapad"},156,{"s":null,"i":"YmKkte_1\/C5DOxd"},["AimExportActionToastLink",309275]]--><\/div>\n<p><!--TgQPHd|[[null,0,null,0,null,0]]--><\/p>\n<\/div>\n<p><!--TgQPHd|[]--><\/p>\n<\/div>\n<p><!--TgQPHd|[]--><\/p>\n<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-mcsf=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"tonYlb\" data-tr-rsts=\"false\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 24px 0px 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"ilZyRc R7mRQb\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-animation-skip=\"true\" aria-hidden=\"false\" data-ved=\"0CAAQ_sYPahcKEwjIiK7d_NOUAxUAAAAAHQAAAAAQWA\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"xEFZqe\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px 0px 0px 184.656px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"tbIZh wQN2Jd Odbbif\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"sUKAcb\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"VndcI veK2kb\" role=\"heading\" aria-level=\"2\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"iMqumd\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: -1px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">You said: <\/span><span data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">I don’t understand this identity:… Note: W(y)\/y = e^[-W(y)] …<\/span><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"UYpEO\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 12px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(86, 89, 94);\">\u00a0<\/div>\n<div class=\"dcCF7d\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<\/div>\n<\/div>\n<div class=\"OBWDNe\" data-ved=\"0CAEQ3PkPahcKEwjIiK7d_NOUAxUAAAAAHQAAAAAQWA\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"\" data-streaming-container=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-container-id=\"OwwUaoTDCcjvkdUP1Na7wAY\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"CKgc1d\" data-scope-id=\"turn\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-subtree=\"aimc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"Zkbeff\" data-sfc-root=\"c\" data-wiz-uids=\"EEbXsd_3\" data-sfc-cb=\"\" data-ved=\"2ahUKEwjEzIa9htSUAxXId6QEHVTrDmgQ2O0OegYIAAgBEAA\" data-hveid=\"CAAIARAA\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"pWvJNd\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\u00a0<\/div>\n<div class=\"mZJni Dn7Fzd\" dir=\"ltr\" data-container-id=\"main-col\" data-xid=\"VpUvz\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-ved=\"2ahUKEwjEzIa9htSUAxXId6QEHVTrDmgQ3KYQegYIAAgBEAE\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIAhAA\" data-processed=\"true\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">This identity looks confusing at first, but it is actually a direct rearrangement of the <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">definitional property<!--TgQPHd|[]--><\/strong> you looked at earlier: \\(W(y)e^{W(y)} = y\\).<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIAxAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Here is the quick, 3-step proof showing exactly where it comes from:<!--TgQPHd|[]--><\/div>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div class=\"otQkpb\" role=\"heading\" aria-level=\"3\" data-animation-nesting=\"\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-processed=\"true\" data-sae=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 20px; font-weight: 600; margin: 24px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">The 3-Step Derivation<!--TgQPHd|[]--><\/div>\n<ol class=\"IaGLZe VimKh\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li style=\"list-style-type: none;\">\n<ol class=\"IaGLZe VimKh\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBRAA\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Start with the definition:<!--TgQPHd|[]--><\/strong><br data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\" \/>\\(W(y)e^{W(y)}=y\\)<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBRAC\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Divide both sides by \\(y\\):<!--TgQPHd|[]--><\/strong><br data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\" \/>\\(\\frac{W(y)e^{W(y)}}{y}=1\\)<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<li class=\"Z1qcYe\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBRAF\" data-complete=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><span class=\"T286Pc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Divide both sides by \\(e^{W(y)}\\) to isolate the fraction:<!--TgQPHd|[]--><\/strong><br data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\" \/>\\(\\frac{W(y)}{y}=\\frac{1}{e^{W(y)}}\\)<!--TgQPHd|[]--><\/span><!--TgQPHd|[]--><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<p><!--TgQPHd|[]--><\/p>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBhAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Using the rule of exponents where \\(\\frac{1}{e^A} = e^{-A}\\), the right side becomes \\(e^{-W(y)}\\).<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIBxAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">This leaves you with the exact identity:<br data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\" \/>\\(\\frac{W(y)}{y}=e^{-W(y)}\\)<!--TgQPHd|[]--><\/div>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<hr class=\"j3tEEe\" data-sfc-root=\"c\" data-sfc-pl=\"|[]\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 32px 0px; text-decoration: none; border-bottom: 0px rgb(128, 128, 128);\" \/>\n<div class=\"Fsg96\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 14px; font-weight: 400; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\"><!--TgQPHd|[]--><\/div>\n<div class=\"otQkpb\" role=\"heading\" aria-level=\"3\" data-animation-nesting=\"\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-sae=\"\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 20px; font-weight: 600; margin: 24px 0px 12px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Why is this identity useful?<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIChAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">In calculus and algebra, this identity is a powerful tool used to <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">move a variable out of a denominator<!--TgQPHd|[]--><\/strong>.<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAICxAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">If you are simplifying an expression and you see \\(\\frac{1}{y}\\) multiplied by \\(W(y)\\), you can instantly swap it out for a clean exponential term \\(e^{-W(y)}\\), which is often much easier to integrate or differentiate.<!--TgQPHd|[]--><\/div>\n<div class=\"n6owBd awi2gc\" data-sfc-cp=\"\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-hveid=\"CAAIDBAA\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 400; margin: 12px 0px 16px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">Would you like to see how this specific identity changes the way you take the <strong class=\"Yjhzub\" data-sfc-root=\"c\" data-sfc-cb=\"\" data-complete=\"true\" data-processed=\"true\" data-copy-service-computed-style=\"font-family: "Google Sans", Arial, sans-serif; font-size: 16px; font-weight: 600; margin: 0px; text-decoration: none; border-bottom: 0px rgb(10, 10, 10);\">derivative<!--TgQPHd|[]--><\/strong> of the Lambert \\(W\\) function?<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Solving 5^x = 100x – 175 Using Lambert W Function Solving 5x = 100x \u2212 175 Using the Lambert W Function Objective: Solve the equation systematically using the Lambert W method, without guessing solutions. 1Rewrite the Equation We start with: 5x = 100x \u2212 175 Using the exponential form 5x = ex ln 5: ex […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_themeisle_gutenberg_block_has_review":false,"footnotes":""},"class_list":["post-45","page","type-page","status-publish","hentry"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.9 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Using the Lambert W Function - Math Lab<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/potentsky.com\/math\/using-the-lambert-w-function\/\" \/>\n<meta property=\"og:locale\" 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guessing solutions. 1Rewrite the Equation We start with: 5x = 100x \u2212 175 Using the exponential form 5x = ex ln 5: ex […]","og_url":"https:\/\/potentsky.com\/math\/using-the-lambert-w-function\/","og_site_name":"Math Lab","article_modified_time":"2026-05-25T09:48:27+00:00","twitter_card":"summary_large_image","twitter_misc":{"Est. reading time":"7 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/potentsky.com\/math\/using-the-lambert-w-function\/","url":"https:\/\/potentsky.com\/math\/using-the-lambert-w-function\/","name":"Using the Lambert W Function - Math 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