Why does $x^{\log_a y} = y^{\log_a x}$ ? (intuitive reason)

Question

So I have seen mathematical proofs for this. They used other logarithmic identities to get this result. I am looking for a more intuitive approach to this and not just equations which lead to this.

edit: okay so what I mean by intuitive is that, x^{power of a which gives y} = y^{power of a which gives x}. So is there a way to make sense of it using pure logic and not identities?

Answer 1

$$ (a^m)^n=(a^n)^m,$$

where $a^m$ is $x$ and $a^n$ is $y$.

Answer 2

By definition

$$Y=\log_a y \iff a^Y =y \iff y^\frac 1 Y=a$$ $$X=\log_a x \iff a^X =x\iff x^\frac 1 X=a$$

therefore

$$x^\frac 1 X=y^\frac 1 Y \iff x^Y=y^X$$

or in other words

$$x^{\log_a y} = y^{\log_a x} \iff x^{\frac1{\log_a x}} = y^{\frac1{\log_a y}} \iff a=a$$