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, log_{10}5, log_{10}7 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 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 × 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 $\dfrac{i \pi}{2}$.

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**FAQs**

**FAQs**

**Q1: What are common logarithms?**

Answer: Common logarithms are the logarithms with base 10. For example, log_{10}2 is a common logarithm.

**Q2: ****What are natural logarithms?**

**What are natural logarithms?**

Answer: Natural logarithms are the logarithms with base e. For example, log_{e}2 is a natural logarithm.