# Solving Logarithmic Equations

Standard

While reading John Derbyshire’s Prime Obsession I came across the following statement (clearly explained on pp. 74):

Any positive power of $\log(x)$ eventually increases more slowly than any positive power of $x$.

It is easy to prove this (existence) analytically, by taking derivative to compare slopes. But algebraically it implies that (for example):

There are either no real solution or two real solutions of the equation
$\log(x) = x^\varepsilon$
for any given $\varepsilon>0$.

Now the question that arises is “How to find this $x$?” I had no idea about how to solve such logarithmic equations, so I took help of Google and discovered this Mathematic.SE post. So, we put $\log(x)=y$ and re-write the equation as:

$y=e^{y\varepsilon}$

Now to be able to use Lambert W function (also called the product logarithm function) we need to re-write the above equation, but I have failed to do so.

But using WolframAlpha I was able to solve $\log(x)=x^2$ to get $x=e^{\frac{-W(-2)}{2}}$ (which is an imaginary number, i.e. no real solution of this equation) but I was not able to figure out the steps involved. So if you have any idea about the general method or the special case of higher exponents, please let me know.