How do I prove $\log(x^n)=n\log|x|$?

Question

By definition we know that: $\log(x^n)=n\log|x|$ as known property in logarithm function .

If it's not a trivial question, how do I prove that :$\log(x^n)=n\log|x|$?

Note: $x$ is real number, $n$ is a natural number.

Thank you for any help.

Answer 1

It is only true when $x^n> 0$, so we assume it.

We'll use the following definition, which is how Wikipedia and Wolfram define it: $$\log_b x=k\iff b^k=x$$

together with the exponentiation rule: $\,\displaystyle{b^{xy}=\left(b^y\right)^x}$

$$\log_b(x^n)=n\log_b |x|\iff b^{n\log_b |x|}=x^n$$

$$\iff \left(b^{\log_b |x|}\right)^n=x^n\iff |x|^n=x^n$$

$$\iff |x^n|=x^n,$$

which is true.

Answer 2

Consider $log(x) : (0,\infty ) \to (-\infty,\infty )$

Defined by $x=e^{log(x)}$

Consider $n \in \mathbb{N}$

$x^n=e^y$

$\iff log(x^n)=y$

$\iff x^n=e^y$

$\iff x=e^{y/n}$

$\iff log(x)=y/n$

$\iff nlog(x)=y$

Answer 3

There are already correct answers above/below, but for clarity sake, I try to put another (a more clear?) version: Let $x \neq 0$ and suppose $y = \log (x^n)$ (here unless we are in complex domain, $x^n >0$). Then by definition $ e^y = x^n$. Since $e^y$ is positive $$ |x| = e^{y/n} $$ (this is analytically true). Taking back $\log$ we get $$ n \log(|x|) = y $$

Thus you have proved a formula:

Prop For $x \in \mathbb{R} , n \in \mathbb{N}$ such that $x^n >0$, $$ n \log(|x|) = \log(x^n). $$