# How do you calculate log_5 33 with a calculator?

Dec 12, 2016

${\log}_{5} 33 = 2.1724$

#### Explanation:

The easiest way to solve such problems is to first change it to base $10$, as tables for base $10$ are easily available. For this we can use the identity ${\log}_{b} a = \log \frac{a}{\log} b$, where $\log a$ and $\log b$ are to base $10$ (if base is not mentioned, it is taken as $10$ and if base is $e$, we use $\ln$ - for natural logarithm).

Here, we have ${\log}_{5} 33$ and it is equal to $\log \frac{33}{\log} 5$ and from tables this is equal to

$\frac{1.5185}{0.6990}$

= $2.1724$

For using scientific calculator just press $5$ and then button $\log$, which gives $0.6990$ (up to four places of decimals), now put this number in memory. Now press $33$ and then button $\log$, which gives $1.5185$ up to four places of decimals. Now divide by memory recall (which has $0.6990$ in memory) to get answer.