It's an intricate lattice that preservers a huge amount of structure — all the structure of the numbers, in fact, given that $~$\\log$~$ is invertible\. $~$\\log\_2(3)$~$ is a number that is simultaneously $~$1$~$ less than $~$\\log\_2(6),$~$ and half of $~$\\log\_2(9)$~$, and a tenth of $~$\\log\_2(3^{10}),$~$ and $~$\\log\_2(3^9)$~$ less than $~$\\log\_2(3^{10}).$~$ The value of $~$\\log\_2(3)$~$ has to satisfy a massive number of constraints, in order to be precisely the number such that multiplication on the left corresponds to addition on the right\. It's no surprise, then, that it's transcendental\.
Wait, really? Is this a joke or does being transcendental follow from having to satisfy many constraints?