Answer
$2\log x + \frac{3}{2} \log (1-5x) -\frac{1}{2}\log x-\frac{1}{2}\log (x-1)-\frac{1}{2}\log (x+1)$
Work Step by Step
Expanding logarithm: $\log \left(\frac{x^2(1-5x)^{3/2}}{\sqrt {x^3-x}}\right),$
$\log(x^2(1-5x)^{3/2})-\log (\sqrt {x^3-x}),$
$\log x^2 +\log (1-5x)^{3/2}-\log (\sqrt {x^3-x}),$
$2\log x + \frac{3}{2} \log (1-5x) -\frac{1}{2}\log (x^3-x)$
$2\log x + \frac{3}{2} \log (1-5x) -\frac{1}{2}\log x(x-1)(x+1)$
$2\log x + \frac{3}{2} \log (1-5x) -\frac{1}{2}\log x-\frac{1}{2}\log (x-1)-\frac{1}{2}\log (x+1)$
