Solving $x \log(x)$ = $100$.

Question

Is there a way I could solve $x \log(x)$ = $100$?

Answer 1

There is a solution to the equation $$E: x\ln x = 100$$ given by $$x=e^{W(100)}$$ where $W$ is the Lambert W function.

For a generalisation of this type of problem, see this Wiki section on how to solve equations of the form $$x\log_b(x) =a$$

Here, you may note that $b=e$ and $a=100$. Then, the equation becomes: $$x\ln x = 100$$ Then use the method in Example 2 in the same page to get the required answer.

If you want the approximate value of $x$, you may ask WolframAlpha.

Answer 2

$$x=e^{W(100)}$$ and see here: https://en.wikipedia.org/wiki/Lambert_W_function

Answer 3

You can obtain many useful information about the solution from the following basic facts about $f(x)=x\log x$

• $f(x)$ is continuos and defined for $x>0$
• $f(x)\le0$ for $0<x\le 1$
• $f(x)\to +\infty$ as $x\to +\infty$

thus at least one solution exists for some $x>1$.

Since

• $f'(x)=1+\log x>0$ for $x>1$

$\implies f(x)$ is strictly increasing and the solution is unique.

Those simply considerations should be always done before to proceed with any calculations in order to understand "where" look for a solution and and how many slutions we are going to find.

As already pointed out in other answers, the solution can't be find in closed form by elementary function, you can solve by special functions (i.e. Lambert's function) or by numerical methods.

Answer 4

$$x=\frac{100}{\log(x)}$$ so $$x=\frac{100}{\log\frac{100}{\log\frac{100}{\log\frac{100}{\log...}}}}=29.53659905...$$