# What is log5? How can we find logs of numbers without using a calculator?

Jun 8, 2016

$\log 5 = 0.6990$

#### Explanation:

Easiest way is to calculate $\log 5$ by referring to logarithmic tables, which shows $\log 5 = 0.6990$

Another way could be using $\log 2 = 0.3010$ (again for this we need log tables) as $\log 5 = \log \left(\frac{10}{2}\right) = \log 10 - \log 2 = 1 - 0.3010 = 0.6990$

In fact one need to remember log (to the base $10$) for first ten numbers and it makes things lot easier. Note that while $\log 1 = 0$, $\log 10 = 1$. In fact, you just need to know $\log 2 = 0.3010$, $\log 3 = 0.4771$ and $\log 7 = 0.8451$ and then all logs can be worked out using these as

$\log 4 = 2 \log 2$,

$\log 5 = 1 - \log 2$,

$\log 6 = \log 2 + \log 3$,

$\log 8 = 3 \log 2$ and

$\log 9 = 2 \log 3$

If you want to go further, you need to remember $\log 11 = 1.0414$, $\log 13 = 1.1139$, $\log 17 = 1.2304$ and $\log 19 = 1.2788$, rest can be easily calculated. For example $\log 14 = \log 2 + \log 7 = 1.1461$.