Negative of Logarithm of x plus Root x squared plus a squared/Corollary
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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:
\(\ds -\map \ln {x + \sqrt {x^2 - a^2} }\) | \(=\) | \(\ds \map \ln {-x + \sqrt {x^2 + a^2} } - \map \ln {a^2}\) | Negative of Logarithm of x plus Root x squared plus a squared | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -\map \ln {x + \sqrt {x^2 - a^2} }\) | \(=\) | \(\ds \ln \size {x - \sqrt {x^2 + a^2} } - \map \ln {a^2}\) |
$\blacksquare$