Chapter 4, Exponential and Logarithmic Functions - Section 4.4 - Laws of Logarithms - 4.4 Exercises - Page 395: 71

$\ln(x+\sqrt{x^{2}-1})$

$-\ln(x-\sqrt{x^{2}-1})=\ln(x-\sqrt{x^{2}-1})^{-1}=\ln(\frac{1}{x-\sqrt{x^{2}-1}})=\ln(\frac{1}{x-\sqrt{x^{2}-1}}*\frac{x+\sqrt{x^{2}-1}}{x+\sqrt{x^{2}-1}})=\ln(\frac{x+\sqrt{x^{2}-1}}{x^{2}-(x^{2}-1)})=\ln(\frac{x+\sqrt{x^{2}-1}}{1})=\ln(x+\sqrt{x^{2}-1})$