Logarithm power law discrepancy

Question

According to the power law:-

$$\log_a (x^k) =k\log_a x $$

So take the following example:-

$$\log_2 ((-2)^2) $$

On solving $\log_2 4=2$

However, if we use the power law, then on simplifying, $2\log_2 (-2)$ is not defined.

So how do I justify this?

Answer 1

This dicrepancy is due to the fact that

• $\log x^2$ is defined for $x\neq 0$ but
• $2\log x$ is defined for $x>0$

then the two expression are equal $\iff x>0$.

What is true $\forall x\neq 0$ is that

• $\log x^2=2\log |x|$

where we have used that $\sqrt {x^2}=|x|$.

Answer 2

The power law is

$$\forall x>0:\log x^k=k\log x.$$

There is no power law for the negatives.