Answer
$\displaystyle \ln\frac{x^{2}}{(x^{2}-1)^{2}}$
Work Step by Step
See Th.5.2.
Property $1$ : $\ln(1)=0$
Property $2$ : $\ln(ab)=\ln a + \ln b$
Property $3$ : $\ln(a^{n})=n\cdot\ln a $
Property 4 : $\displaystyle \ln(\frac{a}{b})=\ln a - \ln b$
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$2[\ln x-\ln(x+1)-\ln(x-1)]$
$=2[\ln x-(\ln(x+1)+\ln(x-1))]$= ... property $2$...
$=2\{\ln x-\ln[(x+1)(x-1)]\}$= ... property $4$...
$=2\displaystyle \ln(\frac{x}{(x+1)(x-1)})$= ... property $3$...
$=\displaystyle \ln(\frac{x}{(x+1)(x-1)})^{2}$
$=\displaystyle \ln\frac{x^{2}}{(x+1)^{2}(x-1)^{2}}$ or (recognize a difference of squares)...
$=\displaystyle \ln\frac{x^{2}}{(x^{2}-1)^{2}}$
