# Inverse Hyperbolic Cosine of x over a in Logarithm Form

## Theorem

$\cosh^{-1} \dfrac x a = \ln \left({x + \sqrt {x^2 - a^2} }\right) - \ln a$

## Proof

 $\displaystyle \cosh^{-1} \frac x a$ $=$ $\displaystyle \ln \left({\frac x a + \sqrt{\left({\frac x a}\right)^2 - 1} }\right)$ Definition of Inverse Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \ln \left({\frac x a + \sqrt{\frac {x^2 - a^2} {a^2} } }\right)$ $\displaystyle$ $=$ $\displaystyle \ln \left({\frac x a + \frac {\sqrt{x^2 - a^2} } a}\right)$ $\displaystyle$ $=$ $\displaystyle \ln \left({\frac {x + \sqrt{x^2 - a^2} } a}\right)$ $\displaystyle$ $=$ $\displaystyle \ln \left({x + \sqrt{x^2 - a^2} }\right) - \ln a$ Difference of Logarithms

$\blacksquare$