constant to the $\log(n)$ equals $n$ to $\log(\text{constant})$

Question

Could someone explain to me why

$$x^{\log(n)} = n^{\log(x)}$$ in simple terms?

I tried to simply take the $log$ of both sides but it doesn't work out or simplify.

Answer 1

Recall that logs and exponential functions are inverses, so they cancel, giving the following equality:

$$10^{\log(a)}=a$$

Using this, one can see that

$$x^{\log(n)}=(10^{\log(x)})^{\log(n)}=10^{\log(x)\log(n)}=(10^{\log(n)})^{\log(x)}=n^{\log(x)}$$

Answer 2

Remember the property $$\log(x^a) = a \log x \qquad a > 0$$ So taking $\log$ on both sides yields $$\log(x^{\log n}) = \log(n^{\log x}) \quad \Leftrightarrow\quad \log n \log x = \log x \log n$$ if $x,n \neq 1$, $x>0$. Now the equality should be more obvious.