What logarithm law is applied in saying $3^{\log_{2} (n )} = n^{\log_{2}( 3)}$? Or there tricky math law?

Question

$$3^{\log_{2} (n)} = n^{\log_{2}(3)}$$ How did the magic happen? Or I am just reading it wrong.

Answer 1

It's essentially because of the exponential law $(a^b)^c=a^{bc}$:

$$3^{\log_2(n)}=(2^{\log_2(3)})^{\log_2(n)}=2^{\log_2(3) \cdot \log_2(n)} = (2^{\log_2(n)})^{\log_2(3)}=n^{\log_2(3)}$$

Answer 2

Notice $a^{\log_a b\log_a c} = (a^{\log_a b})^{\log_a c} = b^{\log_a c}$.

But we could just as easily concluded $a^{\log_a b\log_a c} = (a^{\log_a c})^{\log_a b} = c^{\log_a b}$.

So, yep, $b^{\log_a c} = c^{\log_a b}$.