Inverse Hyperbolic Cosine of x over a in Logarithm Form

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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

$\blacksquare$


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