Simple logarithm question

Question

I understand that $\log{b}-\log{c} = \log {(\frac{b}{c})}$

However, why does $a\log{b}-a\log{c} = \log{(\frac{b}{c})}$

Am I misunderstanding something?

Answer 1

$$a\log b - a\log c = a\left(\log (b) - \log (c)\right)= a\log\left(\frac bc\right) = \log\left(\left(\frac bc\right)^a\right)$$