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n
log
2
m
=
m
log
2
n
(1)
n^{\log_2m} = m^{\log_2n} \tag{1}
nlog2m=mlog2n(1)
证明：
n
log
2
m
=
m
log
m
n
log
2
m
=
m
log
2
m
×
log
m
n
n^{\log_2m} = m^{\log_m{n^{\log_2m}}} \\ =m^{\log_2m\times \log_mn}
nlog2m=mlogmnlog2m=mlog2m×logmn
换底公式：
log
m
b
=
log
n
b
log
n
m
\log_mb=\frac{\log_nb}{\log_nm}
logmb=lognmlognb 带入换底公式：
m
log
2
m
×
log
m
n
=
m
log
2
m
×
log
2
n
log
2
m
=
m
log
2
n
m^{\log_2m\times \log_mn} =m^{\log_2m\times \frac{\log_2n}{\log_2m}} \\ =m^{\log_2n}
mlog2m×logmn=mlog2m×log2mlog2n=mlog2n
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