# How do you approximate log_5 (2/3) given log_5 2=0.4307 and log_5 3=0.6826?

Oct 28, 2016

${\log}_{5} \left(\frac{2}{3}\right) \cong - 0.2519$
We can write ${\log}_{5} \left(\frac{2}{3}\right)$ as ${\log}_{5} \left(2\right) - {\log}_{5} \left(3\right)$ using the rule quotient rule of logarithms that ${\log}_{a} \left(\frac{m}{n}\right) = {\log}_{a} \left(m\right) - {\log}_{a} \left(n\right)$.
${\log}_{5} \left(\frac{2}{3}\right) = {\log}_{5} \left(2\right) - {\log}_{5} \left(3\right) = 0.4307 - 0.6826 = - 0.2519$