Prove that $\log_a(1/x)=-\log_a(x)$.

Question

I thought to write $$\log_a(1/x)=\log_a(x^{-1})=-\log_a(x).$$ But it has two problems: when x=0 and on the other problem it doesn't mention any condition. How should I solve it in each of them?

Answer 1

Exclusively by definition:

$$y=\log_a\frac1x\iff a^y=\frac1x=x^{-1}\iff a^{-y}=x\iff -y=\log_ax$$