Where does log(x) / x take maximum value? If the base of the logarithm is `e`, one can say `log(x)/x` takes maximum at `e`. If the base of the logarithm is `10`, one can say `log(x)/x` takes maximum at `10`. But `log10(x)/x` is nothing but `(loge(e)/loge(10))/x`. The two functions are just a constant multiple (`1/loge(10)`) of each other. Shouldn't they have the same maxima?

Your last sentence is correct, whatever base $b$ we take (as long as $b\gt 1)$, the function $\frac{\log_b x}{x}$ reaches a maximum at $x=e$. The reasoning that led to that conclusion is clearly stated.

The assertion in the second sentence that $\frac{\log_{10} x}{x}$ reaches a maximum at $x=10$ is not true.