Question

If g(x) = 8x^6 and f(x) = log4 (2x) then f(g(x)) = ?

Answer 1

$\begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array} ~\hspace{4em} \begin{array}{llll} \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}$

$\begin{cases} g(x)=8x^6\\\\ f(x)=\log_4(2x) \end{cases}\qquad \qquad \begin{array}{llll} f(~~g(x)~~)=\log_4[~2g(x)~] \\\\\\ f(~~g(x)~~)=\log_4[~2(8x^6)~] \end{array} \\\\\\ f(~~g(x)~~)=\log_4(16x^6)\implies f(~~g(x)~~)=\log_4(4^2x^6) \\\\\\ f(~~g(x)~~)=\log_4(4^2)~~ + ~~\log_4(x^6)\implies \boxed{f(~~g(x)~~)=2~~ + ~~6\log_4(x)}$