Answer
$\displaystyle \ln\left(\frac{x^{2}+1}{x^{2}-1}\right)^{3/2}$
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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$\displaystyle \frac{3}{2}[\ln(x^{2}+1)-\ln(x+1)-\ln(x-1)]$
$=\displaystyle \frac{3}{2}\{\ln(x^{2}+1)-[\ln(x+1)+\ln(x-1)]\}$= ... property $2$...
$=\displaystyle \frac{3}{2}\{\ln(x^{2}+1)-\ln[(x+1)(x-1)]\}$= ... property $4$...
(also, recognize a difference of squares...)
$=\displaystyle \frac{3}{2}\cdot\ln\frac{x^{2}+1}{x^{2}-1}$= ... property $3$...
$\displaystyle \ln\left(\frac{x^{2}+1}{x^{2}-1}\right)^{3/2}$
