Negative of Logarithm of x plus Root x squared plus a squared/Corollary

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

Let $x > 1$.

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

Proof
We have that $\sqrt {x^2 + a^2} > x$ for all $x$.

Hence for all $x$:
 * $-x + \sqrt {x^2 + a^2} > 0$

and so:
 * $x - \sqrt {x^2 + a^2} < 0$

Hence:
 * $-x + \sqrt {x^2 + a^2} = \size {x - \sqrt {x^2 + a^2} }$

Then we have:

Also see

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