From the Introduction to logarithm, we know that the value of a logarithm is meaningless if we do not mention the base.

Definition of Common Logarithm 

The logarithm with base 10 is called the common logarithm. This is also known as the decimal logarithm.

For example,  log₁₀5, log₁₀7 are the common logarithms of 5 and 7 respectively.

Solved Problems on Common Logarithm 

Ex 1: Calculate the common logarithm of 10.

Solution: 

The common logarithm of 10 is denoted by log₁₀ 10

As we know that log_a a=1, we have log₁₀ 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 log₁₀ 100

By the power rule of logarithm $[\log_a b^n=n \log_a b]$,

we get that

log₁₀ 100 = log₁₀ 10² = 2 log₁₀ 10 =2 × 1 = 2

Thus the common logarithm of 100 is 2.

Next, we discuss the natural logarithm.

Definition of Natural Logarithm 

The logarithm with base $e$ is called the natural logarithm. Here  $e$ is the well-known irrational number whose value is

$e=\sum_{n=0}^\infty \dfrac{1}{n!}$.

The natural logarithm of $M$ is usually denoted as $\ln M$. Note that  $\log M \neq \ln M$.

Solved Problems on Natural Logarithm 

Ex 1: 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 2: Calculate 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 $-1=e^{i \pi}$

∴ $\log_e (-1)=\log_e e^{i \pi}$

⇒ $\log_e(-1)=\log_e e^{i \pi}$

⇒ $\log_e(-1)=i \pi \quad$ [∵ log_a a^b = b]

So $ \ln(-1)=i \pi$.

Thus the natural logarithm of -1 is iπ.

That is, ln(-1) = iπ

Ex 3: 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 $i=\sqrt{-1}$

∴ $\log_e i=\log_e \sqrt{-1}$

⇒ $\log_e i=\log_e (-1)^{1/2}$

⇒ $\log_e i=\dfrac{1}{2} \log_e (-1) \quad$  [∵ log_a x^b = b log_a x]

⇒ $\log_e i=\dfrac{1}{2} \ln (-1)$

⇒ $\log_e i=\dfrac{1}{2} i \pi \quad$ [ by Ex 4]

So $ \ln i=\dfrac{i \pi}{2}$.

Thus the natural logarithm of $i$ is $\dfrac{i \pi}{2}$.

ALSO READ

• Logarithm: Definition, Examples, Properties, Solved Problems
• Logarithm Formulas with Proofs

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