冪指数として対数を持つ場合

$a^{{\log }_{b}c}=c^{{\log }_{b}a}$

$$\begin{array}{rcl}& & a^{{\log }_{b}c}\\ & =& a^{\frac{{\log }_{a}c}{{\log }_{a}b}}& \cdots & {\log }_{A}B=\frac{{\log }_{C}A}{{\log }_{C}B}(\mathrm{底}\mathrm{の}\mathrm{変}\mathrm{換})\\ & =& {\left(a^{{\log }_{a}c}\right)}^{\frac{1}{{\log }_{a}b}}& \cdots & A^{BC}={\left(A^B\right)}^C\\ & =& c^{\frac{1}{{\log }_{a}b}}& \cdots & A^{{\log }_{A}B}=B\\ & =& c^{{\log }_{b}a}& \cdots & a^{\frac{1}{{\log }_{b}c}}=a^{{\log }_{c}b}\end{array}$$