# Common Logarithm and Natural Logarithm

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 loga 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 $-1=e^{i \pi}$

∴ $\log_e (-1)=\log_e e^{i \pi}$

⇒ $\log_e(-1)=\log_e e^{i \pi}$

⇒ $\log_e(-1)=i \pi \quad$ [loga ab = 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 $i=\sqrt{-1}$

∴ $\log_e i=\log_e \sqrt{-1}$

⇒ $\log_e i=\log_e (-1)^{1/2}$

⇒ $\log_e i=\dfrac{1}{2} \log_e (-1) \quad$  [ loga xb = b loga 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 $\dfrac{i \pi}{2}$.

# FAQs

### Q1: What are common logarithms?

Answer: Common logarithms are the logarithms with base 10. For example, log102 is a common logarithm.

### Q2: What are natural logarithms?

Answer: Natural logarithms are the logarithms with base e. For example, loge2 is a natural logarithm.