I'm struggling with this one. Assuming the natural log, Mathematica gives the following result: $$-\log\left(\log\left(\frac{2}{\pi^2}\right)+2\log(\pi)-\log(2)\right)=\infty$$ But working it out myself I see that: $$\log\left(\frac{2}{\pi^2}\right)+2\log(\pi)-\log(2) = 0$$ Which simplifies to:$$-\log(0)$$ Is this undefined? Is the original equation malformed?
Usually, I can make a mental picture to understand this stuff but I'm struggling here. I sense a fracturing, perhaps akin to what happens in rocks during an earthquake, or when quaternions go rogue in an otherwise decent subroutine.
Curiously, WolframAlpha gives the following transcendental result: $$-\log\left(\log\left(\frac{2}{\pi^2}\right)+2\log(\pi)-\log(2)\right)=50\log(2)$$
I feel like I'm missing something really obvious but I can't quite sniff it out.
Calculate the middle part first. Apparently $\log(\frac{2}{\pi^2})+2\log(\pi)-\log(2)=\log(2)-\log(2)=0$ from logarithmic identities.
The final result, $-\log(0)$, is undefined since $0$ is a singularity point for the logarithmic function. We may nevertheless interpret the Mathematica result as a limit, as $\displaystyle\lim_{x\rightarrow0}-\log(x)=+\infty$.