Don't understand how powers of logarithm work.

Question

I found this example and I'm struggling to derive it so that the left hand side is equal to right hand side. Why is this so?

$$5^{\log_2(x)} = x^{\log_2(5)}$$

Answer 1

$$ 5^{\log_2(x)} = 5^{\color{red}{1} \cdot \log_2(x)} = 5^{\color{red}{\tfrac{\log_2(5)}{\log_2(5)}} \cdot \log_2(x)} = 5^{\color{blue}{\tfrac{\log_2(x)}{\log_2(5)}} \cdot \log_2(5)} = 5^{\color{blue}{\log_5(x)} \cdot \log_2(5)} = x^{\log_2(5)} $$

Answer 2

The numbers $2$ and $5$ are irrelevant here. If $a,b>0$, then$$a^b=e^{b\log a}$$and therefore, if $c>0$\begin{align}a^{\log_cb}&=e^{\log_c(b)\log(a)}\\&=e^{\frac{\log b}{\log c}\log a}\\&=e^{\log b\frac{\log a}{\log c}}\\&=b^{\log_ca}.\end{align}

Answer 3

To prove $5^{\log_2 x}=x^{\log_2 5}$, apply $\log_2$ to $5^{\log_2 x}$:

$\log_2 5^{\log_2 x} = \log_2 x\log_2 5 = \log_2 5 \log_2 x = \log_2 x^{\log_2 5}.$

You just need one property of the logarithm ($\log a^b = b\log a$) and commutativity ($ab=ba$) in the 2nd equation.

Then exponentiate both sides (which are equal) by 2:

$2^{\log_2 5^{\log_2 x}} = 5^{\log_2 x}$ and $2^{\log_2 x^{\log_2 5}} = x^{\log_2 5}$.

Here you just need that $2^{\log_2 y} = y$, i.e., exponentiation by 2 and taking $\log_2$ are inverse operations.