From the introduction to logarithm, we know that the value of a logarithm is meaningless if we do not mention the base.
Common Logarithm:
The logarithm with base 10 is called the common logarithm. This is also known as decimal logarithm.
For example, `log_{10} 5,` `log_{10} 7` are the common logarithms of 5 and 7 respectively.
Solved Problems:
Ex 1: Calculate the common logarithm of 10.
Solution:
The common logarithm of 10 is denoted by `log_{10} 10`
As we know that `log_a a=1`, we have `log_{10} 10=1`.
Thus the common logarithm of 10 is 1.
Ex 2: Find the common logarithm of 100.
Solution:
The common logarithm of 100 is denoted by `log_{10} 100`
By the power rule of logarithm `[log_a b^n=n log_a b]`,
we get that
`log_{10} 100=log_{10} 10^2=2 log_{10}{10=2 cdot 1=2`
Thus the common logarithm of 100 is 2.
Next, we discuss the natural logarithm.
Natural Logarithm:
The logarithm with base `e` is called the natural logarithm. Here `e` is the well-known irrational number whose value is
`e=sum_{n=0}^infty frac{1}{n!}`.
The natural logarithm of `M` is usually denoted as `ln M`. Note that `log Mneq ln M`.
Solved Problems:
Ex 3: Calculate the natural logarithm of `e`.
Solution:
We have to calculate `ln(e)`
We know that the natural logarithm has base `e`
Thus, `ln(e)=log_e e`
As `log_a a=1`, we deduce that `log_e e=1`.
Thus the natural logarithm of `e` is `1`.
That is, `ln(e)=1`
Ex 4: Find the natural logarithm of `-1`.
Solution:
We have to calculate `ln(-1)`
As the natural logarithm has base `e`, we have
`ln(-1)=log_e (-1)`
We know that `-1=e^{i pi}`
`therefore log_e (-1)=log_e e^{i pi}`
`Rightarrow log_e(-1)=log_e e^{i pi}`
`Rightarrow log_e(-1)=i pi quad` `[ because log_a a^b=b]`
So ` ln(-1)=i pi`.
Thus the natural logarithm of `-1` is `i pi`.
That is, `ln(-1)=i pi`
Ex 5: Calculate the natural logarithm of `i`.
Solution:
We have to calculate `ln i`
As the natural logarithm has base `e`, we have
`ln i=log_e i`
We know that `i=sqrt{-1}`
`therefore log_e i=log_e sqrt{-1}`
`Rightarrow log_e i=log_e (-1)^{1/2}`
`Rightarrow log_e i=1/2 log_e (-1) quad` `[ because log_a x^b=blog_a x]`
`Rightarrow log_e i=1/2 ln (-1)`
`Rightarrow log_e i=1/2 i pi quad` [ by Ex 4]
So ` ln i=frac{i pi}{2}`.
Thus the natural logarithm of `i` is `frac{i pi}{2}`.
Also read: