对数互换公式证明

$$n^{{\log }_{2}m}=m^{{\log }_{2}n}\text{\hspace{1em}(1)}$$

证明：
$$\begin{gathered} n^{{\log }_{2}m}=m^{{\log }_m{n}^{{\log }_{2}m}} \\ =m^{{\log }_{2}m\times {\log }_{m}n} \end{gathered}$$

换底公式：$${\log }_{m}b=\frac{{\log }_{n}b}{{\log }_{n}m}$$
带入换底公式：
$$\begin{gathered} m^{{\log }_{2}m\times {\log }_{m}n}=m^{{\log }_{2}m\times \frac{{\log }_{2}n}{{\log }_{2}m}} \\ =m^{{\log }_{2}n} \end{gathered}$$