# How do you calculate log_6 5 with a calculator?

Aug 12, 2016

${\log}_{6} 5 = \log \frac{5}{\log} 6 \approx 0.8982444$

#### Explanation:

Use the change of base formula:

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

So you can use natural or common logs:

${\log}_{6} 5 = \ln \frac{5}{\ln} 6$

${\log}_{6} 5 = \log \frac{5}{\log} 6$

In fact, if you know $\log 2 \approx 0.30103$ and $\log 3 \approx 0.47712$ then you can get a reasonable approximation with basic arithmetic operations:

${\log}_{6} 5 = \log \frac{5}{\log} 6 = \frac{\log 10 - \log 2}{\log 2 + \log 3}$

$\approx \frac{1 - 0.30103}{0.30103 + 0.47712} = \frac{0.69897}{0.77815} \approx 0.89825$

Actually the true value is closer to $0.8982444$