## Calculus 10th Edition

$\displaystyle \frac{1}{2}\ln(x-1)-\frac{1}{2}\ln x$
See Th.5.2. Property $3$ : $\ln(a^{n})=n\cdot\ln a$ Property 4 : $\displaystyle \ln(\frac{a}{b})=\ln a - \ln b$ $\displaystyle \ln\sqrt{\frac{x-1}{x}}=\ln(\frac{x-1}{x})^{1/2}$= ... property $3$... $=\displaystyle \frac{1}{2}\ln(\frac{x-1}{x})$= ... property $4$... $=\displaystyle \frac{1}{2}[\ln(x-1)-\ln x]$ $=\displaystyle \frac{1}{2}\ln(x-1)-\frac{1}{2}\ln x$