Answer
$\dfrac{1}{2}\left[2\ln (5x)+\ln {(1+3x})\right]-3\ln (x-4) $
Work Step by Step
We use the rules of logarithms to obtain:
$\ln \dfrac { 5x\sqrt{( 1+3x)}}{(x-4)^3}=\ln (5x\sqrt {1+3x})-\ln (x-4)^3\\=\ln (\sqrt {(5x)^2(1+3x}))-\ln (x-4)^3 \\=\dfrac{1}{2}\left[2\ln (5x)+\ln {(1+3x})\right]-3\ln (x-4) $
