One of the best mathematical tools ever developed is the logarithm of a number. It has been used extensively for the simplification of lengthy arithmetic calculations. The standard way of using the technique is via tables of common logarithms. The method of using these tables is well known and has been in use for decades.

Today we will talk about a lesser known method of finding log of a number without using the tables. This method has proven to be very useful when a log-table or an electronic calculator is not around. So, here it is...

### Things to memorize

We choose to work with base 10

• $$\log{2}\approx 0.301$$
• $$\log{3} \approx 0.477$$
• $$\log{5} \approx 0.699$$
• $$\log{7} \approx 0.845$$

### Master Formula

The value of $$\log{a}$$ is approximately given by the formula
$$\log{a} \approx \log{b} + \left( \frac{a-b}{b} \right ) 0.4343$$
Here, $$a$$ and $$b$$ are such that $$|a-b|$$ is as small as possible and $b$ is chosen according to the list above. The accuracy of the result depends on smallness of $|a-b|$.

Let us try to evaluate the value of $\log{2.7}$.  So, we have $a = 2.7$ and we choose $b = 2.8$. We can plug these values in the master formula to get

\begin{aligned}\log{2.7} &\approx \log{2.8} + \left( \frac{2.7-2.8}{2.8} \right)0.4343\\ &= \log{\frac{28}{10}} - \frac{0.1}{2.8} \times 0.4343\\&= \log{28} - \log{10} - 0.0155 \\ &= \log{7} +2\log{2} - 1 - 0.0155\end{aligned}

The values of $\log{2}$ and $\log{7}$ are already known(memorized!), so we have

\begin{aligned}\log{2.7} &\approx 0.845 + 2 \times 0.301 - 1.0155\\&=0.4315\end{aligned}

The exact value of $\log{2.7} = 0.4314$ and the error in the result is just 0.03%

For numbers less than $2$, multiply and divide with an appropriate factor to bring it close to 5 or 7 in order to minimize errors. For e.g.

$$\log{1.31} = \log{\left(1.31 \times \frac{4}{4}\right)} = \log{5.24} - \log{4}$$

Now, the master formula can be used as before.

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