# How do you find the exact value of log_5 75-log_5 3?

Jan 7, 2017

${\log}_{5} 75 - {\log}_{5} 3 = 2$

#### Explanation:

Remember that ${\log}_{a} b - {\log}_{a} c = {\log}_{a} \left(\frac{b}{c}\right)$

$\therefore {\log}_{5} 75 - {\log}_{5} 3 = {\log}_{5} \left(\frac{75}{3}\right) = {\log}_{5} 25$

Let $x = {\log}_{5} 25$

$\therefore {5}^{x} = 25$

$\to {5}^{x} = {5}^{2}$

Equating indices:

$x = 2$

Hence: ${\log}_{5} 75 - {\log}_{5} 3 = 2$