Understanding why $x^{\log_b(y)} = y^{\log_b(x)}$ According to wikipedia, we have that 
$$
x^{\log_b(y)} = y^{\log_b(x)} 
$$
because
$$
x^{\log_b(y)} = b^{\log_b(x) \log_b(y)} = b^{\log_b(y) \log_b(x)} = y^{\log_b(x)}
$$
But what justifies that first leap?
$$
x^{\log_b(y)} = b^{\log_b(x) \log_b(y)}
$$

Notice that $\def\lfrac#1#2{{\large\frac{#1}{#2}}}$$a^b=e^{b\ln(a)}$ and $\log_c (d)=\lfrac{\ln(d)}{\ln(c)}$. Therefore:
$$x^{\log_b(y)}=e^{\log_b(y)\ln(x)}=e^{\lfrac{\ln(y)}{\ln(b)}\ln(x)}=e^{\ln(y)\lfrac{\ln(x)}{\ln(b)}}=e^{\ln(y)\log_b(x)}=y^{\log_b(x)}$$
