I am studying maximum likelihood and to simplify taking the derivative of the likelihood function, it is often transformed by the natural log before taking the derivative.
I have read in other posts that this is because the logarithm is a monotonic function, so its extrema will be the same as the original function. However, I do not understand why this is the case. Can someone explain intuitively why the transformation does not affect the local extrema?
Best Answer
Let $f(x)$ be a positive function and suppose that $x_0$ is the local maximum point of $f(x)$ in the interval $[a,b]$. This means that for any $y\in[a,b]$, $f(y)\le f(x_0)$.
Logarithms is a monotonically increasing function: if $z\le w$, then $\log z\le \log w$. So for any $y\in[a,b]$, since $f(y)\le f(x_0)$, we have $\log f(y) \le \log f(x_0)$. Hence, $x_0$ is also the local maximum point for $\log f(x)$.
The other direction can be proven by noting that the inverse of log is also monotonically increasing.