Prove $\frac{\ln(x)}{x} \le \log_{10} 2$

Question

How would I go about proving the following inequality?

$\frac{\ln(x)}{x} \le \log_{10} 2$

Is there an algebraic solution and if not, why?

Answer 1

$\log_{10}2 = \frac{\ln2}{\ln 10}$. Then, for all positive $x$,

$$ x \leq 2^{\frac{x}{\ln 10}} $$

Then you apply $\ln$ to both sides (given that it's monotonically increasing), and you have the inequality you want.