How are these two exponential logarithmic equations equal?

Question

How does following assertion hold? I have tried some real values but can anyone explain it to me mathematically/algebraically? $$ a^{\log_{b}n} = n^{\log_{b}a} $$

I'm reading something that asserts this, but I do not see the connection. Any help would be appreciated.

Answer 1

taking the logarithm on both sides we have $$\log_b n\ln(a)=\log_b a\ln(n)$$ after this we have $$\frac{\ln(n)}{\ln(b)}\ln(a)=\frac{\ln(a)}{\ln(b)}\cdot \ln(n)$$

Answer 2

Here's a method:

$$\log_ac=\frac{\log_bc}{\log_ba}$$

$$\log_ac\cdot\log_ba=\log_bc\cdot\log_aa$$ $$\log_ac^{\log_ba}=\log_aa^{\log_bc}$$ $$c^{\log_ba}=a^{\log_bc}$$