As I know following statements are correct:
$a^{\log_bc} = (b^{\log_ba})^{\log_bc} = b^{\log_ba\log_bc} = (b^{\log_bc})^{\log_ba} = c^{\log_ba},$
$\log_aa^x = x,$
$\log_ab^c = c\log_ab.$
Then I don't get where is the mistake below:
$ a^{\log_bc} = (\log_mm^a)^{\log_bc} = \log_mm^{a\log_bc} = a\log_bc$
or alternatively
$ a^{\log_bc} = (\log_mm^a)^{\log_bc} = \log_mm^{a\log_bc} = (\log_mm^{\log_bc})^a = \log_bc^a = a\log_bc$
Obviously there is a mistake(s) in above reasoning, but I don't get it.