How do I find a natural log without a calculator?

Hint: These types of problems are pretty straight forward and are pretty easy to solve. We need to have a general basic idea of the topic logarithms and must study the graph of the function as deeply as possible. It is possible to determine the value of the logarithm to some extent without a calculator, but in some special cases, determining the value can be very much difficult. Say for an example, the number whose logarithm is to be calculated is a prime number or a multiple of a prime number (except 2, 3, and 5), then finding the value of such numbers can be cumbersome.

Complete step by step solution:
Now we start off with the solution to the given problem by writing that, suppose we have a number say ‘x’, to find the logarithm of this number, first of all we need to break this number into its factors. Say, \[x=a.b.c.d....\] , where ‘a’, ‘b’, ‘c’….. are the factors of the number ‘x’. Now according to the law of logarithms, we know,
\[\log \left( x \right)=\log \left( a.b.c.d.... \right)=\log \left( a \right)+\log \left( b \right)+\log \left( c \right)+\log \left( d \right)+...\]
Now say that suppose either ‘a’ or ‘b’ or ‘c’ or… is a prime number, then determining its value is impossible. In such cases we will not be able to find the value of the logarithm of the number without a calculator.
We must remember the values of certain logarithm values like,
\[\begin{align}
  & {{\log }_{10}}\left( 2 \right)=0.301 \\
 & {{\log }_{10}}\left( 3 \right)=0.477 \\
 & {{\log }_{10}}\left( 5 \right)=0.699 \\
\end{align}\]

Note: For such types of problems, we must first of all remember to factorise the given number so that the evaluation of the problem becomes very easy. We also need to keep in mind the values of certain logarithms so that we can calculate the value of the logarithm conveniently. If one of the factors of the number inside the logarithm is a prime number then the evaluation of the required problem may be difficult.