Change of Base

For \(a,c>0,\ a,c\ne1\) and \(b>0\),

\(\log_a b=\dfrac{\log_c b}{\log_c a}\)

Example

\(\log_2 5=\dfrac{\log_7 5}{\log_7 2}\)

\(\log_{11}2=\dfrac{\log_p2}{\log_p11}\), where \(p>0,\ p\ne1\).

Take note

For \(x,y>0,\ x,y\ne1\), the reciprocal identity is

\(\log_x y=\dfrac1{\log_y x}\)