# How do you calculate log_3 45 - log_3 9?

Jun 26, 2018

#### Answer:

We will use logarithmic rules to find that the answer is $\log \frac{5}{\log} 3$, or about $1.645$.

#### Explanation:

The first rule we'll use is ${\log}_{a} b - {\log}_{a} c = {\log}_{a} \left(\frac{b}{c}\right)$:

${\log}_{3} 45 - {\log}_{3} 9 = {\log}_{3} \left(\frac{45}{9}\right) = {\log}_{3} 5$

This logarithm doesn't look too friendly. Fortunately, we can use another rule:

${\log}_{b} c = \frac{{\log}_{a} c}{{\log}_{a} b}$

It doesn't matter what base we choose for $a$, as long as each logarithm as the same base:

${\log}_{3} 5 = \frac{{\log}_{10} 5}{{\log}_{10} 3} \approx 1.465$

Hope this helps!