[tex]~~~\ln \left( \sqrt{\dfrac{x-1}x \right )\\\\\\=\ln\left(\dfrac{\sqrt{x-1}}{\sqrt x} \right)\\\\\\=\ln\left(\sqrt{x-1} \right) - \ln \left( \sqrt x \right)~~~~~~~~~~~~~~~~~~~~;\left[\log_b\left( \dfrac mn \right) = \log_b m - \log_b n \right]\\\\\\=\ln\left[\left(x -1\right)^{\tfrac 12} \right] - \ln\left(x^{\tfrac 12} \right) \right)\\\\\\=\dfrac 12 \ln(x-1) - \dfrac 12 \ln x~~~~~~~~~~~~~~~~~~~~~~~~;[\log_b m^n = n \log_b m][/tex]
