Is $\lim_{b\to\infty} (x\log_b(x)) = 0$?

Question

Is $$\lim_{b\to\infty} (x\log_b(x)) = 0$$?

I started to attempt this problem by graphing $y=x\log_2(x)$ and kept increasing the base of log to see what's the behaviour.

You can see here that as the base becomes very very large, the curve tends to straighten out but still it isnt straightening out enough to say that it would become a straight line i.e y = 0.

Is that equation true? How do go on about it?

Answer 1

We can use that $\log_AB=\frac{\log B}{\log A}$ therefore

$$x\log_bx=x\frac{\log x}{\log b}\to 0$$

Answer 2

We can use the fact of logarithms that as $\log_BA = \frac{log A}{log B}$

then expression $\lim_{b \to \infty} x\log_b(x)$ can be rewritten as

$\lim_{b \to \infty} \frac{x\log x}{log b}$ and as $b \to \infty$ then $\log b \to \infty$ and hence making $\frac{1}{\log b} \to 0$ and in

$\lim_{b \to \infty} \frac{x\log x}{log b}$ b is the variable which is changing and hence making $x$ as constant and also $x\log x$ as

constant, so $\lim_{b \to \infty} x\log_b(x) = 0$