Let's say that $a^x>a^y$ and $a>1$. You take the $\log_a$ of both functions and you get $x>y$.
What if I wanted to do the same with $a^x+a^y>a^z$? Then I would get $\log_a(a^x+a^y)>z$.
I wonder if $\log_a(a^x+a^y)$ can be shortened? Is there a rule that I am not aware of, or can I not do anything else? I am only wondering this in form of algebra, not computing. Thanks in advance!
The logarithm of a sum can only be expressed with infinitely many terms. Thus, you cannot shorten (by which I take you to mean reduce the amount of symbols in your sum) significantly. You may only lengthen it to infinity, if you please.