How do I find the common logarithm of a number?

See the explanation.
If you have technology available for the logarithm in some other base ( ${\displaystyle e}$ or ${\displaystyle 2}$ ), use
${\displaystyle {\log }_{10}n=\frac{{\log }_{b}n}{{\log }_b\left\{10\right\}}(\text{ where }b=e\text{ or }2)}$
With paper and pencil, I don't know a good series for ${\displaystyle {\log }_{10}n}$.
Probably the simplest way is to use a series for ${\displaystyle \ln n}$ and either a series or memorization for ${\displaystyle \ln 10\approx 2.302585093}$
For ${\displaystyle \ln n}$ , let ${\displaystyle x=n-1}$ and use:
${\displaystyle \ln n=\ln (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots }$
After you find ${\displaystyle \ln n}$ , use division to get ${\displaystyle {\log }_{10}n\approx \frac{\ln n}{2.302585093}}$

Hint: First we should understand the term ‘logarithm’ and then we will see two types of logarithm in mathematics and understand the term ‘common logarithm’. Then we will consider some examples like ${\displaystyle {10}^2,{10}^3}$ , etc. and find the value of their common logarithm.
Complete step by step answer:
In mathematics, the logarithm is the inverse function of exponentiation. That means that the logarithm of a given number ‘n’ is the exponent to which another fixed number the base ‘b’ must be raised to produce that number ‘n’. Mathematically, we denote it as ${\displaystyle {\log }_{b}n}$, here ‘n’ is called theargument and ‘b’ is called the base. Let us assume ${\displaystyle {\log }_{b}n=x}$ , this means that b is raised to the power x to get the value n. ${\displaystyle {\log }_{b}n=x}$ can be written in exponential form as ${\displaystyle b^x=n}$ .
Now, there are two types of logarithm namely: common logarithm and natural logarithm. The main difference between these logarithms is that in the common log we take 10 as the value of the base while in the natural log we take e as the value of base, where ${\displaystyle e\approx 2.71}$ . In the above question, we have to consider the common logarithm only. Let us take an example, we consider a number 100. Let us find its common log. So, we have, ${\displaystyle {\log }_{10}100}$ . That means we have to determine the number x which when raised as a power of 10 gives 100. We know that 10 raised to the power 2 is 100. So, we get, ${\displaystyle \Rightarrow {\log }_{10}100=2}$
Similarly, ${\displaystyle {\log }_{10}1000=3}$