Solving 5^x = 100x − 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:

Using the exponential form 5^x = e^{x ln 5}:

2Eliminate the Constant Term in the Linear Part

We want the right-hand side to be of the form α(x − β). Factor 100:

So the equation becomes:

3Shift the Variable

Let t = x − 7 4 , so x = t + 7 4 .

Substitute into the equation:

Let k = 100 5^7/4 , so:

4Rearrange into Lambert W Form

Divide both sides by k t and rearrange:

Multiply both sides by −ln 5:

Let u = −t ln 5, then:

5Apply Lambert W Function

The Lambert W function is defined as the inverse of f(w) = w e^w. Therefore:

Since x = t + ⁷⁄₄:

6Simplify the Constant Inside W

Recall k = ¹⁰⁰⁄_5^7/4. We need to simplify:

Note that 5^7/4 = 5^2−1/4 = 25 · 5^−1/4. Therefore:

Let A = ln 5 4 · 5^1/4 . Then:

7Find W(−A) for Both Real Branches

Numerically:

So −A ≈ −0.269, which is greater than −¹⁄_e ≈ −0.3679.

8Solve for W(−A) Exactly

We need W(−A) such that:

First Branch (W₀):

Try W₀(−A) = − ln 5 4

Check:

Therefore: W₀(−A) = − ln 5 4

Second Branch (W₋₁):

Try W₋₁(−A) = − 5 ln 5 4

Check:

Therefore: W₋₁(−A) = − 5 ln 5 4

9Final Solutions

Verification

This solution demonstrates the power of the Lambert W function in solving transcendental equations that combine exponential and linear terms.