Why is $\arg\max \space f(x) = \arg\min \space \left\{-\log \space f(x) \right\}$?

Question

Why is

$$\arg\max \space f(x) = \arg\min \space \left\{-\log \space f(x) \right\}$$

The left side is the $x$ value where the function $f(x)$ has the maximum value. The right side is confusing me. The questions comes from the logistic regression.

Answer 1

Note that $\log_a(x)$ is a strictly monotonic increasing function for any $a>1$. Therefore, its negative is a decreasing function, so $-\log f(x)$ is minimized iff $f(x)$ is maximized, and vice versa.