Negative of Logarithm of x plus Root x squared minus a squared

Theorem
Let $x \in \R: \size x > 1$.

Let $x > 1$.

Then:
 * $-\map \ln {x + \sqrt {x^2 - a^2} } = \map \ln {x - \sqrt {x^2 - a^2} } - \map \ln {a^2}$

Proof
First we note that if $x > 1$ then $x + \sqrt {x^2 - a^2} > 0$.

Hence $\map \ln {x + \sqrt {x^2 - a^2} }$ is defined.

Then we have:

Also see

 * Negative of Logarithm of x plus Root x squared plus a squared