# 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, log_{G}N = a

N = G^{a}

To help you understand the concept of logarithms more easily, you need to return to indices. Logarithms and Indices are closely related. You can read up about indices here.

From indices, we know that 3^{2 }= 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:

Log_{3}9 = 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:

- Product Law
- Quotient law
- Raising to a Power
- 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:

Log_{a} (GH) = Log_{a }G + Log_{a} H

Consider,

Log_{a }G = v ; G = a^{v }and

Log_{a }H = y ; H = a^{y}

GH = a^{v }x a^{y }

= a^{v+y} (first law of indices)

Then, Log_{a} (GH) = v + y

= Log_{a }G + Log_{a} H

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

Given that 4 = 2^{2}, then Log_{2 }4 = 2 and

If 8 = 2^{3}, then Log_{2}8 = 3

Then 4 x 8 will be given as:

4 x 8 = 2^{2 }x 2^{3}

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

= 2^{2+3} = 2^{5}

Therefore; Log_{2 }(4 x 8) = 2 + 3

= Log_{2 }4 + Log_{2 }8

**Worked Examples**

Evaluate the following given that log_{10 }3 = 0.4771, log_{10 }2 = 0.3010, and log_{10} 7 = 0.8451. (i) log_{10 }6 (ii) log_{10 }42

**Solutions**

- Log
_{10 }6 = log_{10}(2 x 3)

Log_{10 }2 + log_{10 }3

= _{ }0.3010 + 0.4771

0.7781

(ii) log_{10 }42 = log_{10}(7 x 6)

Log_{10 }(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:

Log_{a }(G ÷ H) = Log_{a} G – Log_{a }H

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

Given that, Log_{a} G = x, G = a^{x }and Log_{a} H = y, H = a^{y}

_{ }(G ÷ H) = a^{x} – a^{y} = a^{x-y}

Then, Log_{a }(G ÷ H) = x – y = Log_{a} G – Log_{a }H

**Worked Examples**

Evaluate the following given that log_{10 }3 = 0.4771, log_{10 }2 = 0.3010, and log_{10} 7 = 0.8451.

- log
_{10 }(3/2) - Find log
_{10 }(7/2) - log
_{10 }(6/2)

**Solutions**

- log
_{10 }(3/2) = log_{10}3 – log_{10 }2

0.4771- 0.3010

= 0.1761

- log
_{10 }(7/2) = log_{10}7 – log_{10 }2

0.8451 – 0.3010

= 0.5441

- log
_{10 }(6/2) = log_{10}( 3 x 2) – log_{10 }2

= (log_{10} 3 + log_{10 }2) – log_{10 }2

(0.4771+ 0.3010) – 0.3010

0.7781 – 0.3010

= 0.4771

**Raising To A Power**

By mathematical definition, this law states:

Log_{a }M^{p} = p log_{a }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 n^{th} root of a number, G is the logarithm of the number divided by n”.

Mathematically,

Log ^{n}√G = log G ÷ n

**Worked Example**

Solve log^{3}√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.

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