Juxtaposing Logs

$\log_a(b^c) = \log_c(b^a) \\ c\log_ab = a\log_cb\\ \dfrac{c}{a} = \dfrac{\log_cb}{\log_ab} \\ = \dfrac{1}{\log_ab} \times \dfrac{1}{\log_bc} \\ = \dfrac{1}{\dfrac{1}{\log_ba}} \times \dfrac{1}{\log_bc} \\ = \dfrac{\log_ba}{\log_bc} \\ = \log_ca \\ \dfrac{c}{a} = \log_ca \\ c = a\log_ca \\ c = \log_c(a^a) \boxed{\therefore c^c = a^a}$

$\log_a b^c = \log_c b^a \\ \implies c \log_a b = a \log_c b \\ \implies c \cdot \dfrac{\cancel{\log b}}{\log a} = a \cdot \dfrac{\cancel{\log b}}{\log c} \\ \implies \dfrac{c}{\log a} = \dfrac{a}{\log c} \\ \implies c \log c = a \log a \\ \implies \log c^c = \log a^a \\ \implies \boxed{c^c = a^a}$