If $p$ is negative, is it true that $\ln(p^2) = 2\ln(p)$?

Question

Suppose, $p$ is a real negative number. However, $p^2$ is positive. Now,

$$\ln(p^2) = 2 \ln(p)\tag{1}$$

Question:

Probably not valid in your context. If you start taking logs of negative numbers, you push yourself into complex numbers and then $\ln$ could have multiple values. You have to choose a "branch" of $\ln$ and then you still have to be careful. — B. Goddard, Oct 12 '21 at 11:26

score 5 · Accepted Answer · edited Oct 12 '21 at 20:04

5

No!

For exactly the reason you mention.

However, for $p\in\mathbb{R}$, $p\neq 0$, it is correct to write $\ln(p^2) = 2\ln(|p|)$.

edited Oct 12 '21 at 20:04

RiaD

answered Oct 12 '21 at 11:25

G. Fougeron

1 Answers1