Is $\log(1+x)\sim x$ correct? This is my problem if $\log(1+x)\sim x$ when we are given $x\to 0$?

The assignment doesn't mention if $\log$ is $\log_{10}$ or $\log_{e}$? I assumed the first version and saw the claim is wrong. Am I on a right track?
I would definitely go with the taylor expansion around the origin:

$$\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\mathcal O(n^4)\simeq x \ \ for \ |x|\ll 1$$
$$\lim_{x \to 0} \log_b(1+x) = \lim_{x \to 0}\log_b(1) = 0$$

Because $b^0 = 1 \space \forall b$