Power and base rules

Question

We know $2^{xy} = 2^x + 2^y$. Why can't we apply it to logs. E.g.

$e^{x\log y}= e^{x} + e^{\log y}$

Answer 1

It seems like you are confusing the rules of exponents. Your first statement: $2^{xy} = 2^x+2^y$ is generally not true (Ex: $2^{1\cdot 1} \neq2^1 + 2^1$). However, $2^{x+y} = 2^x\cdot 2^y$ is true, even when $x$ and $y$ are logarithmic functions. As a commenter mentioned, trying to prove these identities may help you understand them a little better. Hope this helps!