Common Logarithm and Natural Logarithm: Definition, Examples, Difference | Log vs Ln

From the introduction to logarithm, we know that the value of a logarithm does not make any sense without the base. In the topic of logarithms, we often hear the terms common logarithm and natural logarithm. In this section, we will discuss them.

Common Logarithm

The logarithm of a number with base 10 is called the common logarithm of that number. It is also known as decimal logarithm.

For example, the logarithm of 7 with base 10, that is, log₁₀ 7 is called the common logarithm of 10.

Note: The common logarithm of a number M is usually denoted as $\log M.$ So both log₁₀M and $\log M$ have the same meaning. In other words, both represent the same number.

Natural Logarithm

Note: The natural logarithm of a number $x$ is usually denoted as $\ln x.$ From the above discussion, we see that the numbers $\log x$ and $\ln x$ are different.

Log vs Ln

The difference between log and ln is provided in the table below.

Solved Examples of common logarithms:

Solution: 

We need to find log₁₀ 10

As log_aa=1, we have log₁₀10=1.

So the common logarithm of 10 is 1.

Solution:

Note that 1000=10³.

So log₁₀ 1000 = log₁₀10³ = 3

$[\because \log_a a^n=n]$

So the common logarithm of 1000 is 3.

Solved Examples of natural logarithms:

Solution: 

We need to find log_ee

As log_aa =1, we have log_e e=1.

So the natural logarithm of e is 1.

Solution:

As the natural logarithm has base e, we have to find log_e (-1)

It is known that -1 = e^iπ

So log_e (-1) = log_e e^iπ = iπ

$[\because \log_a a^n=n]$

So the natural logarithm of -1 is iπ.

Solution:

We need to find log_e i

We known that i = $e^{i \frac{\pi}{2}}$

So log_e i = log_e $e^{i \frac{\pi}{2}}$ = iπ/2.

$[\because \log_a a^n=n]$

So the natural logarithm of i is iπ/2.

ALSO READ:

• An Introduction to Logarithm
• Logarithm formulas or Logarithm rules
• Solved problems of logarithms

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