Real Area Hyperbolic Cosine of x over a in Logarithm Form

Theorem

$\arcosh \dfrac x a = \map \ln {x + \sqrt {x^2 - a^2} } - \ln a$

$\ds \arcosh \frac x a$

$=$

$\ds \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} }$

$\ds $

$=$

$\ds \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } }$

$\ds $

$=$

$\ds \map \ln {\frac x a + \frac {\sqrt {x^2 - a^2} } a}$

$\ds $

$=$

$\ds \map \ln {\frac {x + \sqrt {x^2 - a^2} } a}$

$\ds $

$=$

$\ds \map \ln {x + \sqrt {x^2 - a^2} } - \ln a$

$\blacksquare$