# Logarithm ‖ Definition, Laws And Worked Examples

This article will provide a detailed explanation of logarithms and their associated laws with relevant work examples. It will also provide a step–by–step methodological approach to solving each arithmetic problem involving a logarithm. Ensure that you read to the end.

## Definition Of Logarithm

When you hear the word Logarithm, the first thing that should come to your mind is exponent or power. The Logarithm of a number N to any base G is the index or power to which the base G must be raised to equal the number N.

Mathematically, if a is the logarithm of a number N to base G,

Then, logGN = a

N = Ga

From indices, we know that 32 = 9, where 3 is the base, and 2 is the power or index.

We can easily represent this in the logarithm as the log of 9 to base 3 equals 2.

Mathematically denoted as:

Log39 = 2

## Laws Of Logarithm

The laws of logarithms are rules you must observe to solve arithmetic problems involving logarithms correctly. We are going to list some basic laws of logarithms with worked examples to help you understand each law. It is important to note that some of the laws of logarithms share some similarities with the laws of indices.

The laws of logarithm are:

1. Product Law
2. Quotient law
3. Raising to a Power
4. Roots law

### Product Law

You can define the logarithm of a product as the sum of the logarithms of the factors that make up the product.

This law can be defined mathematically as follows:

Loga (GH) = Loga G + Log­a H

Consider,

Loga G = v ; G = av and

Loga H = y ; H = ay

GH = av x ay

= av+y (first law of indices)

Then, Loga (GH) = v + y

= Loga G + Log­a H

To bring more clarity on the product law to you, let’s consider the example below:

Given that 4 = 22, then Log2 4 = 2 and

If 8 = 23, then Log28 = 3

Then 4 x 8 will be given as:

4 x 8 = 22 x 23

By the first law of indices, we sum up the power

= 22+3 = 25

Therefore; Log2 (4 x 8) = 2 + 3

= Log2 4 + Log2 8

#### Worked Examples

Evaluate the following given that log10 3 = 0.4771, log10  2 = 0.3010, and log10 7 = 0.8451. (i) log10 6 (ii) log10 42

##### Solutions
• Log10 6 = log10 (2 x 3)

Log10 2 + log10 3

=  0.3010 + 0.4771

0.7781

(ii)     log10 42 = log10(7 x 6)

Log10 (7 x 2 x 3)

0.3010 + 0.4771 + 0.8451

= 1.6232

### Quotient Law

The logarithm of a quotient is the difference between the logarithm of the dividend and the divisor. Simply put, the quotient law is the opposite of the product law.

Mathematically, quotient law can be expressed as follows:

Loga (G ÷ H) = Loga G – Loga H

You can derive this law easily from the second law of indices.

Given that, Loga G = x, G = ax and Loga H = y, H = ay

(G ÷ H) = ax – ay = ax-y

Then, Loga (G ÷ H) = x – y = Loga G – Loga H

#### Worked Examples

Evaluate the following given that log10 3 = 0.4771, log10  2 = 0.3010, and log10 7 = 0.8451.

• log10 (3/2)
• Find log10 (7/2)
• log10 (6/2)
##### Solutions
• log10 (3/2) = log10 3 – log10 2

0.4771- 0.3010

= 0.1761

• log10 (7/2) = log10 7 – log10 2

0.8451 – 0.3010

= 0.5441

• log10 (6/2) = log10 ( 3 x 2) – log10 2

= (log10 3 + log10 2) – log10 2

(0.4771+ 0.3010) – 0.3010

0.7781 – 0.3010

=  0.4771

### Raising To A Power

By mathematical definition, this law states:

Loga Mp = p loga M

#### Worked Examples

Evaluate the following:

• log 10 (100)2
• log 10 (1000)2
##### Solutions
• log 10 (100)2 = 2 log 10 (100)
• x 2 = 4
• log 10 (1000)2 = 2 log 10 (1000)

2 x 3 = 6

### Roots Law

The roots law states “the logarithm of the nth root of a number, G is the logarithm of the number divided by n”.

Mathematically,

Log n√G = log G ÷ n

#### Worked Example

Solve log3√1000

(Log 1000) ÷ 3

3 ÷ 3 =1

You can solve simple and complex problems involving logarithms with all the above explanations and worked examples. Please use the comment section below if you have any questions or comments.