# How do you use the Change of Base Formula and a calculator to evaluate the logarithm log_3 40?

Jul 26, 2015

You divide $\log \left(40\right)$ by $\log \left(3\right)$ or divide $\ln \left(40\right)$ by $\ln \left(3\right)$

#### Explanation:

Change of base means that you divide the log (to the new base) of the number by the log (to the new base) of the old base.

You may use $\log = {\log}_{10}$ or the $\ln = {\log}_{e}$ as long as you use the same button both times.

So:
${\log}_{3} 40 = \log \frac{40}{\log} 3 = \frac{1.6020 \ldots}{0.4771 \ldots} = 3.3577 \ldots$
Or:
${\log}_{3} 40 = \ln \frac{40}{\ln} 3 = \frac{3.6888 \ldots}{1.0986 \ldots} = 3.3577 \ldots$

Of course you can do this in one go, as long as you mind the brackets.