TITLE: Is $\ln(x)$ ever greater than $x$?
QUESTION [8 upvotes]: Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement?
Just wondering for some convergence related thing

REPLY [5 votes]: We can get a stronger result using the method shown by @johannesvalks. Let
$$ f(x) = x - e \ln x. $$
Then
$$ f'(x) = 1 - \frac{e}{x}. $$
The function $f(x)$ has a local maximum or minimum only when $f'(x) = 0$, namely at
$$ x = e. $$
We can show this is a minimum either by taking the second derivative or by examining
$f(x)$ at some other positive value of $x$.  Therefore, for all $x > 0$,
$$ f(x) = x - e \ln x \ge f(e) = 0. $$
That is,
$$ x \ge e \ln x. $$
You can use this fact to prove other things such as your statement in a comment
that $(log_{10} x)^4 < x$.