Is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true?

Question

Assuming

$\mathbf{x}\in \mathbb{R}^n$,

$f(\mathbf{x})\gt0 \forall\mathbf{x}\in\mathbb{R}^n$,

is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true?

Why?

Answer 1

due to $\log$ is monotonous and $f(x) \geq f(y)$ if and only if $\log f(x) \geq \log f(y)$