# How do you evaluate log_3 64?

Dec 26, 2016

I found $3.78557$

#### Explanation:

I would try to change base and use a pocket calculator. The natural log, $\ln$, can normally be evaluated using a pocket calculator so I 'll try to use $\ln$:
${\log}_{3} \left(64\right) = \frac{\ln \left(64\right)}{\ln \left(3\right)} = 3.78557$

Remember that to change base to a new base $c$ you do:
${\log}_{b} \left(a\right) = \frac{{\log}_{c} \left(a\right)}{{\log}_{c} \left(b\right)}$

Dec 26, 2016

${\log}_{3} 64 = \frac{6 \log 2}{\log 3} \approx 3.7855786$

#### Explanation:

Suppose you know the following approximations:

${\log}_{10} 2 \approx 0.30103$

${\log}_{10} 3 \approx 0.47712125$

The change of base formula tells us that:

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

for any $a , b , c > 0$ with $a , c \ne 1$

So we find:

${\log}_{3} 64 = \frac{{\log}_{10} 64}{{\log}_{10} 3} = \frac{{\log}_{10} {2}^{6}}{{\log}_{10} 3} = \frac{6 {\log}_{10} 2}{{\log}_{10} 3} \approx \frac{6 \cdot 0.30103}{0.47712125} \approx 3.7855786$