Class 9 – Logarithms

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  • Introduction to logarithms
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  • Laws of logarithms
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Introduction: Logarithms were invented independently by John Napier, a Scotsman, and by Joost Burgi, a Swiss. Napier’s logarithms were published in 1614; Burgi’s logarithms were published in 1620. The objective of both men was to simplify mathematical calculations. This approach originally arose out of a desire to simplify multiplication and division to the level of addition and subtraction.

Of course, in this era of the cheap hand calculator, this is not necessary anymore but it still serves as a useful way to introduce logarithms. Napier’s approach was algebraic and Burgi’s approach was geometric. The invention of the common system of logarithms is due to the combined effort of Napier and Henry Biggs in 1624. Natural logarithms first arose as more or less accidental variations of Napier’s original logarithms. Their real significance was not recognized until later. The earliest natural logarithms occur in 1618.

Definition: If ‘a’ is a positive real number (a ≠ 1) and N is a positive real number there exists a real number ‘x’ such that then ‘x’ is the logarithm of ‘N’ to the base a and we write it as


i) , the logarithm of 36 to the base 6 is ‘2’ i.e



But (-3)2 = 9 cannot be written as
= 2

is called exponential form and is called logarithm form. The logarithm of N to the base a is usually written as . Note that the same meaning is expressed by two equations.

Find the logarithm of to the base


Thenby definition of logarithm

Equatingpowers on both sides


1. In loga N, neither N nor a is negative but the value of loga N can be negative.

2. If loga N ≠ loga M thenN ≠ M i.e., logarithm of different numbers to the same base are different. If .

3. Logarithm of same number to different bases aredifferent i.e. .


Propertiesof Logarithms.

1. Logarithm of a positive real number to the same base is unity.

Proof: Let a be a positive real number(a ≠ 1)

We have .

2. Logarithm of unity of any positivereal base (≠1) is zero.

Proof: Let a be any positive real number (a≠ 1)

We have



(logarithmic form)

Remove log with base a on both sides

Hence proved. Logarithm of zero is undefined

Signof logarithm: (Under various cases)

Sign of a

Sign of N

Sign of Logarithm


i) a > 1

N > 1

ii) a > 1

0 < N < 1

iii) 0 < a < 1

N > 1

iv) 0 < a < 1

0 < N < 1

1. Find x by converting each of the following to exponential form.
a) b)


2. a) If , express in terms of x.
b) If , and then express z in terms of x and y.



1. If a and N have the same parity then is positive otherwise negative

2. If a > 1, N > M _
In this case is said to be an increasing function .

3. If 0 < a < 1; N > M _

In this case is said to be a decreasing function.