Question

How do you solve `2^x = 10x` ?

Answer 1

There is no pure algebraic solution using elementary functions, but there is an effective numerical method.

Explanation:

Given:

`2^x = 10x`

Define:

`f(x) = 2^x - 10x`

Differentiating, we get:

`f'(x) = 2^x ln 2 - 10`

Using Newton's method, we can choose an initial approximation `a_0`, then derive successively better approximations by applying the formula:

`a_(i+1) = a_i - (f(a_i))/(f'(a_i)) = a_i - (2^(a_i)-10a_i)/(2^(a_i)ln 2-10)`

Putting this into a spreadsheet with `a_0 = 1`, I got the following:

Then setting `a_0 = 5`, I got this:

So there seem to be two solutions:

`x_1 ~~ 0.1077550150002717`

`x_2 ~~ 5.8770105937921375`

Looking at the graphs of `2^x` and `10x` we have:

graph{(y-2^x)(y-10x) = 0 [-7, 13, -11.1, 68.9]}