冪指数として対数を持つ場合

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

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

冪指数として対数の逆数を持つ場合

$a^{\frac{1}{{\log }_{b}c}}=a^{{\log }_{c}b}$

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