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 5x<\/i><\/sup> = 100x<\/i> \u2212 175 Using the Lambert W Function<\/h1>\n \n \n Objective:<\/strong> Solve the equation systematically using the Lambert W method, without guessing solutions.\n <\/p>\n\n <h2>1<\/span>Rewrite the Equation<\/h2>\n We start with:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 5x<\/i><\/sup> = 100x<\/i> \u2212 175\n <\/div>\n <\/div>\n Using the exponential form 5x<\/i><\/sup> = e<\/i>x<\/i> ln 5<\/sup>:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n e<\/i>x<\/i> ln 5<\/sup> = 100x<\/i> \u2212 175\n <\/div>\n <\/div>\n\n <h2>2<\/span>Eliminate the Constant Term in the Linear Part<\/h2>\n We want the right-hand side to be of the form \u03b1<\/i>(x<\/i> \u2212 \u03b2<\/i>). Factor 100:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 100x<\/i> \u2212 175 = 100(x<\/i> \u2212 \n 175<\/span>\n 100<\/span>\n <\/span>)<\/span> = 100(x<\/i> \u2212 \n 7<\/span>\n 4<\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n So the equation becomes:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n e<\/i>x<\/i> ln 5<\/sup> = 100(x<\/i> \u2212 \n 7<\/span>\n 4<\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n\n <h2>3<\/span>Shift the Variable<\/h2>\n Let t<\/i> = x<\/i> \u2212 \n 7<\/span>\n 4<\/span>\n <\/span><\/span>, so x<\/i> = t<\/i> + \n 7<\/span>\n 4<\/span>\n <\/span><\/span>.<\/p>\n \n Substitute into the equation:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n e<\/i>(t<\/i> + \n 7<\/span>\n 4<\/span>\n <\/span>) ln 5<\/sup> = 100t<\/i>\n <\/div>\n <div class=\"equation\">\n e<\/i>t<\/i> ln 5<\/sup> \u00b7 e<\/i>\n 7<\/span>\n 4<\/span>\n <\/span> ln 5<\/sup> = 100t<\/i>\n <\/div>\n <div class=\"equation\">\n 5\n 7<\/span>\n 4<\/span>\n <\/span><\/sup> \u00b7 e<\/i>t<\/i> ln 5<\/sup> = 100t<\/i>\n <\/div>\n <div class=\"equation\">\n e<\/i>t<\/i> ln 5<\/sup> = \n 100<\/span>\n 57\/4<\/sup><\/span>\n <\/span>t<\/i>\n <\/div>\n <\/div>\n \n Let k<\/i> = \n 100<\/span>\n 57\/4<\/sup><\/span>\n <\/span><\/span>, so:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\">\n e<\/i>t<\/i> ln 5<\/sup> = k t<\/i>\n <\/div>\n <\/div>\n\n <h2>4<\/span>Rearrange into Lambert W Form<\/h2>\n Divide both sides by k t<\/i> and rearrange:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 1 = k t e<\/i>\u2212t<\/i> ln 5<\/sup>\n <\/div>\n <div class=\"equation\">\n \n 1<\/span>\n k<\/i><\/span>\n <\/span> = t e<\/i>\u2212t<\/i> ln 5<\/sup>\n <\/div>\n <\/div>\n \n Multiply both sides by \u2212ln 5:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span> = (\u2212t<\/i> ln 5) e<\/i>\u2212t<\/i> ln 5<\/sup>\n <\/div>\n <\/div>\n \n Let u<\/i> = \u2212t<\/i> ln 5, then:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\">\n \u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span> = u eu<\/sup><\/i>\n <\/div>\n <\/div>\n\n <h2>5<\/span>Apply Lambert W Function<\/h2>\n The Lambert W function is defined as the inverse of f<\/i>(w<\/i>) = w ew<\/sup><\/i>. Therefore:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n u<\/i> = W<\/i>(\u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <div class=\"equation\">\n t<\/i> = \u2212\n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 W<\/i>(\u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n \n Since x<\/i> = t<\/i> + 7<\/sup>\u20444<\/sub>:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\" style=\"font-size: 18px;\">\n x<\/i> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 W<\/i>(\u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n\n <h2>6<\/span>Simplify the Constant Inside W<\/h2>\n Recall k<\/i> = 100<\/sup>\u204457\/4<\/sup><\/sub>. We need to simplify:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span> = \u2212\n ln 5 \u00b7 57\/4<\/sup><\/span>\n 100<\/span>\n <\/span>\n <\/div>\n <\/div>\n \n Note that 57\/4<\/sup> = 52\u22121\/4<\/sup> = 25 \u00b7 5\u22121\/4<\/sup>. Therefore:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \n ln 5 \u00b7 57\/4<\/sup><\/span>\n 100<\/span>\n <\/span> = \n ln 5 \u00b7 25 \u00b7 5\u22121\/4<\/sup><\/span>\n 100<\/span>\n <\/span> = \n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span>\n <\/div>\n <\/div>\n \n Let A<\/i> = \n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span><\/span>. Then:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\" style=\"font-size: 18px;\">\n x<\/i> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 W<\/i>(\u2212A<\/i>)\n <\/div>\n <\/div>\n\n <h2>7<\/span>Find W(\u2212A) for Both Real Branches<\/h2>\n Numerically:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n A<\/i> \u2248 \n 1.609438<\/span>\n 4 \u00d7 1.495349<\/span>\n <\/span> \u2248 \n 1.609438<\/span>\n 5.981396<\/span>\n <\/span> \u2248 0.269\n <\/div>\n <\/div>\n \n So \u2212A<\/i> \u2248 \u22120.269, which is greater than \u22121<\/sup>\u2044e<\/i><\/sub> \u2248 \u22120.3679.<\/p>\n \n <div class=\"note\">\n Important:<\/strong> Since \u2212A<\/i> > \u22121<\/sup>\u2044e<\/i><\/sub>, there are two real branches<\/strong> of the Lambert W function: W<\/i>0<\/sub> (principal branch) and W<\/i>\u22121<\/sub> (secondary branch).\n <\/div>\n\n <h2>8<\/span>Solve for W(\u2212A) Exactly<\/h2>\n \n We need W<\/i>(\u2212A<\/i>) such that:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n W<\/i>(\u2212A<\/i>) \u00b7 e<\/i>W<\/i>(\u2212A<\/i>)<\/sup> = \u2212A<\/i> = \u2212\n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span>\n <\/div>\n <\/div>\n\n <h3>First Branch (W0<\/sub>):<\/h3>\n Try W<\/i>0<\/sub>(\u2212A<\/i>) = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n Check:<\/strong><\/p>\n <div class=\"check-box\">\n <div class=\"equation\">\n (\u2212\n ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> \u00b7 exp(\u2212\n ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5\u22121\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212\n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span> = \u2212A<\/i> \u2713<\/span>\n <\/div>\n <\/div>\n \n Therefore: W<\/i>0<\/sub>(\u2212A<\/i>) = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n <div class=\"solution-box\">\n <div class=\"equation\" style=\"font-size: 20px; color: #f57c00;\">\n x<\/i>1<\/sub> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 (\u2212\n ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \n 7<\/span>\n 4<\/span>\n <\/span> + \n 1<\/span>\n 4<\/span>\n <\/span> = 2<\/span>\n <\/div>\n <\/div>\n\n <h3>Second Branch (W\u22121<\/sub>):<\/h3>\n Try W<\/i>\u22121<\/sub>(\u2212A<\/i>) = \u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n Check:<\/strong><\/p>\n <div class=\"check-box\">\n <div class=\"equation\">\n (\u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> \u00b7 exp(\u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5\u22125\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5 \u00b7 5\u22125\/4<\/sup> = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5\u22121\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212\n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span> = \u2212A<\/i> \u2713<\/span>\n <\/div>\n <\/div>\n \n Therefore: W<\/i>\u22121<\/sub>(\u2212A<\/i>) = \u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n <div class=\"solution-box\">\n <div class=\"equation\" style=\"font-size: 20px; color: #f57c00;\">\n x<\/i>2<\/sub> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 (\u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \n 7<\/span>\n 4<\/span>\n <\/span> + \n 5<\/span>\n 4<\/span>\n <\/span> = 3<\/span>\n <\/div>\n <\/div>\n\n <h2>9<\/span>Final Solutions<\/h2>\n <div class=\"final-answer\">\n x<\/i> = 2 and x<\/i> = 3\n <\/div>\n\n <h2>Verification<\/h2>\n <div class=\"highlight-box\">\n <h3>For x<\/i> = 2:<\/h3>\n <table>\n <tr>\n <td>Left side:<\/strong><\/td>\n <td>52<\/sup> = 25<\/td>\n <\/tr>\n <tr>\n <td>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 x<\/i> = 3:<\/h3>\n <table>\n <tr>\n <td>Left side:<\/strong><\/td>\n <td>53<\/sup> = 125<\/td>\n <\/tr>\n <tr>\n <td>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 \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","protected":false},"excerpt":{"rendered":"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":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","footnotes":""},"class_list":["post-45","page","type-page","status-publish","hentry","entry"],"yoast_head":"\n<title>Using 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{"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":"2025-11-08T14:46:08","modified_gmt":"2025-11-08T14:46:08","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 5x<\/i><\/sup> = 100x<\/i> \u2212 175 Using the Lambert W Function<\/h1>\n \n \n Objective:<\/strong> Solve the equation systematically using the Lambert W method, without guessing solutions.\n <\/p>\n\n <h2>1<\/span>Rewrite the Equation<\/h2>\n We start with:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 5x<\/i><\/sup> = 100x<\/i> \u2212 175\n <\/div>\n <\/div>\n Using the exponential form 5x<\/i><\/sup> = e<\/i>x<\/i> ln 5<\/sup>:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n e<\/i>x<\/i> ln 5<\/sup> = 100x<\/i> \u2212 175\n <\/div>\n <\/div>\n\n <h2>2<\/span>Eliminate the Constant Term in the Linear Part<\/h2>\n We want the right-hand side to be of the form \u03b1<\/i>(x<\/i> \u2212 \u03b2<\/i>). Factor 100:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 100x<\/i> \u2212 175 = 100(x<\/i> \u2212 \n 175<\/span>\n 100<\/span>\n <\/span>)<\/span> = 100(x<\/i> \u2212 \n 7<\/span>\n 4<\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n So the equation becomes:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n e<\/i>x<\/i> ln 5<\/sup> = 100(x<\/i> \u2212 \n 7<\/span>\n 4<\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n\n <h2>3<\/span>Shift the Variable<\/h2>\n Let t<\/i> = x<\/i> \u2212 \n 7<\/span>\n 4<\/span>\n <\/span><\/span>, so x<\/i> = t<\/i> + \n 7<\/span>\n 4<\/span>\n <\/span><\/span>.<\/p>\n \n Substitute into the equation:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n e<\/i>(t<\/i> + \n 7<\/span>\n 4<\/span>\n <\/span>) ln 5<\/sup> = 100t<\/i>\n <\/div>\n <div class=\"equation\">\n e<\/i>t<\/i> ln 5<\/sup> \u00b7 e<\/i>\n 7<\/span>\n 4<\/span>\n <\/span> ln 5<\/sup> = 100t<\/i>\n <\/div>\n <div class=\"equation\">\n 5\n 7<\/span>\n 4<\/span>\n <\/span><\/sup> \u00b7 e<\/i>t<\/i> ln 5<\/sup> = 100t<\/i>\n <\/div>\n <div class=\"equation\">\n e<\/i>t<\/i> ln 5<\/sup> = \n 100<\/span>\n 57\/4<\/sup><\/span>\n <\/span>t<\/i>\n <\/div>\n <\/div>\n \n Let k<\/i> = \n 100<\/span>\n 57\/4<\/sup><\/span>\n <\/span><\/span>, so:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\">\n e<\/i>t<\/i> ln 5<\/sup> = k t<\/i>\n <\/div>\n <\/div>\n\n <h2>4<\/span>Rearrange into Lambert W Form<\/h2>\n Divide both sides by k t<\/i> and rearrange:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n 1 = k t e<\/i>\u2212t<\/i> ln 5<\/sup>\n <\/div>\n <div class=\"equation\">\n \n 1<\/span>\n k<\/i><\/span>\n <\/span> = t e<\/i>\u2212t<\/i> ln 5<\/sup>\n <\/div>\n <\/div>\n \n Multiply both sides by \u2212ln 5:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span> = (\u2212t<\/i> ln 5) e<\/i>\u2212t<\/i> ln 5<\/sup>\n <\/div>\n <\/div>\n \n Let u<\/i> = \u2212t<\/i> ln 5, then:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\">\n \u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span> = u eu<\/sup><\/i>\n <\/div>\n <\/div>\n\n <h2>5<\/span>Apply Lambert W Function<\/h2>\n The Lambert W function is defined as the inverse of f<\/i>(w<\/i>) = w ew<\/sup><\/i>. Therefore:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n u<\/i> = W<\/i>(\u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <div class=\"equation\">\n t<\/i> = \u2212\n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 W<\/i>(\u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n \n Since x<\/i> = t<\/i> + 7<\/sup>\u20444<\/sub>:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\" style=\"font-size: 18px;\">\n x<\/i> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 W<\/i>(\u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span>)<\/span>\n <\/div>\n <\/div>\n\n <h2>6<\/span>Simplify the Constant Inside W<\/h2>\n Recall k<\/i> = 100<\/sup>\u204457\/4<\/sup><\/sub>. We need to simplify:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \u2212\n ln 5<\/span>\n k<\/i><\/span>\n <\/span> = \u2212\n ln 5 \u00b7 57\/4<\/sup><\/span>\n 100<\/span>\n <\/span>\n <\/div>\n <\/div>\n \n Note that 57\/4<\/sup> = 52\u22121\/4<\/sup> = 25 \u00b7 5\u22121\/4<\/sup>. Therefore:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n \n ln 5 \u00b7 57\/4<\/sup><\/span>\n 100<\/span>\n <\/span> = \n ln 5 \u00b7 25 \u00b7 5\u22121\/4<\/sup><\/span>\n 100<\/span>\n <\/span> = \n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span>\n <\/div>\n <\/div>\n \n Let A<\/i> = \n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span><\/span>. Then:<\/p>\n <div class=\"highlight-box\">\n <div class=\"equation\" style=\"font-size: 18px;\">\n x<\/i> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 W<\/i>(\u2212A<\/i>)\n <\/div>\n <\/div>\n\n <h2>7<\/span>Find W(\u2212A) for Both Real Branches<\/h2>\n Numerically:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n A<\/i> \u2248 \n 1.609438<\/span>\n 4 \u00d7 1.495349<\/span>\n <\/span> \u2248 \n 1.609438<\/span>\n 5.981396<\/span>\n <\/span> \u2248 0.269\n <\/div>\n <\/div>\n \n So \u2212A<\/i> \u2248 \u22120.269, which is greater than \u22121<\/sup>\u2044e<\/i><\/sub> \u2248 \u22120.3679.<\/p>\n \n <div class=\"note\">\n Important:<\/strong> Since \u2212A<\/i> > \u22121<\/sup>\u2044e<\/i><\/sub>, there are two real branches<\/strong> of the Lambert W function: W<\/i>0<\/sub> (principal branch) and W<\/i>\u22121<\/sub> (secondary branch).\n <\/div>\n\n <h2>8<\/span>Solve for W(\u2212A) Exactly<\/h2>\n \n We need W<\/i>(\u2212A<\/i>) such that:<\/p>\n <div class=\"equation-block\">\n <div class=\"equation\">\n W<\/i>(\u2212A<\/i>) \u00b7 e<\/i>W<\/i>(\u2212A<\/i>)<\/sup> = \u2212A<\/i> = \u2212\n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span>\n <\/div>\n <\/div>\n\n <h3>First Branch (W0<\/sub>):<\/h3>\n Try W<\/i>0<\/sub>(\u2212A<\/i>) = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n Check:<\/strong><\/p>\n <div class=\"check-box\">\n <div class=\"equation\">\n (\u2212\n ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> \u00b7 exp(\u2212\n ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5\u22121\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212\n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span> = \u2212A<\/i> \u2713<\/span>\n <\/div>\n <\/div>\n \n Therefore: W<\/i>0<\/sub>(\u2212A<\/i>) = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n <div class=\"solution-box\">\n <div class=\"equation\" style=\"font-size: 20px; color: #f57c00;\">\n x<\/i>1<\/sub> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 (\u2212\n ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \n 7<\/span>\n 4<\/span>\n <\/span> + \n 1<\/span>\n 4<\/span>\n <\/span> = 2<\/span>\n <\/div>\n <\/div>\n\n <h3>Second Branch (W\u22121<\/sub>):<\/h3>\n Try W<\/i>\u22121<\/sub>(\u2212A<\/i>) = \u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n Check:<\/strong><\/p>\n <div class=\"check-box\">\n <div class=\"equation\">\n (\u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> \u00b7 exp(\u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5\u22125\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5 \u00b7 5\u22125\/4<\/sup> = \u2212\n ln 5<\/span>\n 4<\/span>\n <\/span> \u00b7 5\u22121\/4<\/sup>\n <\/div>\n <div class=\"equation\">\n = \u2212\n ln 5<\/span>\n 4 \u00b7 51\/4<\/sup><\/span>\n <\/span> = \u2212A<\/i> \u2713<\/span>\n <\/div>\n <\/div>\n \n Therefore: W<\/i>\u22121<\/sub>(\u2212A<\/i>) = \u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span><\/p>\n \n <div class=\"solution-box\">\n <div class=\"equation\" style=\"font-size: 20px; color: #f57c00;\">\n x<\/i>2<\/sub> = \n 7<\/span>\n 4<\/span>\n <\/span> \u2212 \n 1<\/span>\n ln 5<\/span>\n <\/span> \u00b7 (\u2212\n 5 ln 5<\/span>\n 4<\/span>\n <\/span>)<\/span> = \n 7<\/span>\n 4<\/span>\n <\/span> + \n 5<\/span>\n 4<\/span>\n <\/span> = 3<\/span>\n <\/div>\n <\/div>\n\n <h2>9<\/span>Final Solutions<\/h2>\n <div class=\"final-answer\">\n x<\/i> = 2 and x<\/i> = 3\n <\/div>\n\n <h2>Verification<\/h2>\n <div class=\"highlight-box\">\n <h3>For x<\/i> = 2:<\/h3>\n <table>\n <tr>\n <td>Left side:<\/strong><\/td>\n <td>52<\/sup> = 25<\/td>\n <\/tr>\n <tr>\n <td>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 x<\/i> = 3:<\/h3>\n <table>\n <tr>\n <td>Left side:<\/strong><\/td>\n <td>53<\/sup> = 125<\/td>\n <\/tr>\n <tr>\n <td>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 \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","protected":false},"excerpt":{"rendered":"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":{"ocean_post_layout":"","ocean_both_sidebars_style":"","ocean_both_sidebars_content_width":0,"ocean_both_sidebars_sidebars_width":0,"ocean_sidebar":"","ocean_second_sidebar":"","ocean_disable_margins":"enable","ocean_add_body_class":"","ocean_shortcode_before_top_bar":"","ocean_shortcode_after_top_bar":"","ocean_shortcode_before_header":"","ocean_shortcode_after_header":"","ocean_has_shortcode":"","ocean_shortcode_after_title":"","ocean_shortcode_before_footer_widgets":"","ocean_shortcode_after_footer_widgets":"","ocean_shortcode_before_footer_bottom":"","ocean_shortcode_after_footer_bottom":"","ocean_display_top_bar":"default","ocean_display_header":"default","ocean_header_style":"","ocean_center_header_left_menu":"","ocean_custom_header_template":"","ocean_custom_logo":0,"ocean_custom_retina_logo":0,"ocean_custom_logo_max_width":0,"ocean_custom_logo_tablet_max_width":0,"ocean_custom_logo_mobile_max_width":0,"ocean_custom_logo_max_height":0,"ocean_custom_logo_tablet_max_height":0,"ocean_custom_logo_mobile_max_height":0,"ocean_header_custom_menu":"","ocean_menu_typo_font_family":"","ocean_menu_typo_font_subset":"","ocean_menu_typo_font_size":0,"ocean_menu_typo_font_size_tablet":0,"ocean_menu_typo_font_size_mobile":0,"ocean_menu_typo_font_size_unit":"px","ocean_menu_typo_font_weight":"","ocean_menu_typo_font_weight_tablet":"","ocean_menu_typo_font_weight_mobile":"","ocean_menu_typo_transform":"","ocean_menu_typo_transform_tablet":"","ocean_menu_typo_transform_mobile":"","ocean_menu_typo_line_height":0,"ocean_menu_typo_line_height_tablet":0,"ocean_menu_typo_line_height_mobile":0,"ocean_menu_typo_line_height_unit":"","ocean_menu_typo_spacing":0,"ocean_menu_typo_spacing_tablet":0,"ocean_menu_typo_spacing_mobile":0,"ocean_menu_typo_spacing_unit":"","ocean_menu_link_color":"","ocean_menu_link_color_hover":"","ocean_menu_link_color_active":"","ocean_menu_link_background":"","ocean_menu_link_hover_background":"","ocean_menu_link_active_background":"","ocean_menu_social_links_bg":"","ocean_menu_social_hover_links_bg":"","ocean_menu_social_links_color":"","ocean_menu_social_hover_links_color":"","ocean_disable_title":"default","ocean_disable_heading":"default","ocean_post_title":"","ocean_post_subheading":"","ocean_post_title_style":"","ocean_post_title_background_color":"","ocean_post_title_background":0,"ocean_post_title_bg_image_position":"","ocean_post_title_bg_image_attachment":"","ocean_post_title_bg_image_repeat":"","ocean_post_title_bg_image_size":"","ocean_post_title_height":0,"ocean_post_title_bg_overlay":0.5,"ocean_post_title_bg_overlay_color":"","ocean_disable_breadcrumbs":"default","ocean_breadcrumbs_color":"","ocean_breadcrumbs_separator_color":"","ocean_breadcrumbs_links_color":"","ocean_breadcrumbs_links_hover_color":"","ocean_display_footer_widgets":"default","ocean_display_footer_bottom":"default","ocean_custom_footer_template":"","footnotes":""},"class_list":["post-45","page","type-page","status-publish","hentry","entry"],"yoast_head":"\n<title>Using 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