# How do you use the Change of Base Formula and a calculator to evaluate the logarithm log_9 3?

Aug 30, 2015

With a calculator you can use $\log = {\log}_{10}$ or $\ln = {\log}_{e}$ with the change of base formula to find answer $0.5$

Alternatively use ${\log}_{3}$ to find $\frac{1}{2}$ algebraically.

#### Explanation:

The change of base formula tells you that ${\log}_{a} b = \frac{{\log}_{c} b}{{\log}_{c} a}$

So you can use: ${\log}_{9} 3 = \frac{\log 3}{\log 9}$ with common logarithms.

On a calculator this will give:

$\frac{\log 3}{\log 9} \approx \frac{0.4771212547}{0.9542425094} \approx 0.5$

Or you can use ${\log}_{9} 3 = \frac{\ln 3}{\ln 9}$ with natural logarithms.

$\frac{\ln 3}{\ln 9} \approx \frac{1.0986122887}{2.197224577} \approx 0.5$

Alternatively, just do the algebra using ${\log}_{3}$:

${\log}_{9} 3 = \frac{{\log}_{3} 3}{{\log}_{3} 9} = \frac{{\log}_{3} {3}^{1}}{{\log}_{3} {3}^{2}} = \frac{1}{2}$