Derivative of y = $(5x)^{\sin(x)}$

Question

How can I solve this using logarithms?I thought I could break it into $\ln(\sin(x))\cdot(\ln(5)+\ln(x))$? However it seems that it breaks into $\ln(\sin(x))\cdot\ln(5x)$.

Why can I not break up the $\ln(5x)$ into $\ln(5)+\ln(x)$?

Answer 1

Taking logarithms, $\ln(y) = \ln((5x)^{\sin(x)}) = \sin(x) \ln(5 x)$ because (at least when $a > 0$) $\ln(a^b) = b \ln(a)$. And then $\ln(5 x) = \ln(5) + \ln(x)$, so $$ \ln(y) = \sin(x) \ln(5) + \sin(x) \ln(x) $$