How do you calculate ${log}_{6} 5$ $log_6 5$ with a calculator?

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How do you calculate ${log}_{6} 5$ $log_6 5$ with a calculator?

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Answer 1 · 2016-08-12T22:49:58

${log}_{6} 5 = \frac{log 5}{log 6} \approx 0.8982444$ $log_6 5 = log 5 / log 6 ~~ 0.8982444$

Explanation:

Use the change of base formula:

${log}_{a} b = \frac{{log}_{c} b}{{log}_{c} a}$ $log_a b = (log_c b) / (log_c a)$

So you can use natural or common logs:

${log}_{6} 5 = \frac{ln 5}{ln 6}$ $log_6 5 = ln 5 / ln 6$

${log}_{6} 5 = \frac{log 5}{log 6}$ $log_6 5 = log 5 / log 6$

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

${log}_{6} 5 = \frac{log 5}{log 6} = \frac{log 10 - log 2}{log 2 + log 3}$ $log_6 5 = log 5 / log 6 = (log 10 - log 2) / (log 2 + log 3)$

$\approx \frac{1 - 0.30103}{0.30103 + 0.47712} = \frac{0.69897}{0.77815} \approx 0.89825$ $~~ (1-0.30103)/(0.30103+0.47712) = 0.69897/0.77815 ~~ 0.89825$

Actually the true value is closer to $0.8982444$ $0.8982444$

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How do you calculate ${log}_{6} 5$ $log_6 5$ with a calculator?

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