# Logarithm ‖ Definition, Laws And Worked Examples

This article will give you a detailed explanation of logarithms, and the associated laws with relevant work examples. It gives a step–by–step methodological approach to solving each arithmetic problem that involves a logarithm. Ensure to 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 cast your mind back 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 represent this easily in the logarithm as the log of 9 to base 3 is equal to 2.

Mathematically denoted as:

Log_{3}9 = 2

**Laws Of Logarithm**

Laws of logarithm are rules you must observe in order to correctly solve arithmetic problems involving logarithms. We are going to list some basic laws of logarithms with worked examples to help you understand each of the laws accordingly. It is important to note that some of the laws of logarithm 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 of 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

With all the above explanations and worked examples, you will be able to solve simple and complex problems involving logarithms. If you have any questions or comments, please use the comment section below.

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