# How do you show that this statement is true log_5 25=2log_5 5?

Nov 26, 2016

Using log ${a}^{m} \equiv m \log a$ (any log base):

${\log}_{5} 25 = {\log}_{5} {5}^{2}$
$\therefore {\log}_{5} 25 = \left(2\right) {\log}_{5} 5$
$\therefore {\log}_{5} 25 = 2 {\log}_{5} 5$ QED

NB: ${\log}_{a} a = 1$ so we can also write

${\log}_{5} 25 = \left(2\right) \left(1\right)$
$\therefore {\log}_{5} 25 = 2$