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Introduction to

Exponents and Logarithms

Christopher Thomas

Mathematics Learning Centre

University of Sydney

NSW 2006

c ?1991 University of Sydney

Acknowledgements

Parts of section 1 of this booklet rely a great deal on the presentation given in the booklet of the same name, written by Peggy Adamson for the Mathematics Learning Centre in

1987. The remainder is new.

Jackie Nicholas, Sue Gordon and Trudy Weibel read pieces of earlier drafts of this booklet. I should like to thank them for their extremely helpful comments on the contents and layout. In addition, Duncan Turpie performed the laborious task of final proof reading.

Thanks Duncan.

Christopher Thomas

December 1991

This booklet was revised in 1998 by Jackie Nicholas.

Contents

1 Exponents 1

1.1 Introduction . .................................. 1

1.2 Exponents with the Same Base........................ 1

1.3 Exponents with Different Bases........................ 7

1.4 Scientific Notation . .............................. 8

1.5 Summary . . .................................. 9

1.6 Exercises..................................... 10

2 Exponential Functions 11

2.1 The Functionsy=2

x andy=2 -x ....................... 11

2.2 The functionsy=b

x andy=b -x ....................... 12

2.3 The Functionsy=e

x andy=e -x ....................... 13

2.4 Summary . . .................................. 14

2.5 Exercises..................................... 14

3 Logarithms 15

3.1 Introduction . .................................. 15

3.2 Logarithms to Base 10 (Common Logarithms)................ 15

3.3 Logarithms to Base b .............................. 20

3.4 Logarithms to Basee(Natural Logarithms) ................. 22

3.5 Exponential Functions Revisited........................ 22

3.6 Summary . . .................................. 23

3.7 Exercises..................................... 23

4 Solutions to Exercises 24

4.1 Solutions to Exercises from Section 1 ..................... 24

4.2 Solutions to Exercises from Section 2 ..................... 27

4.3 Solutions to Exercises from Section 3. ..................... 29

Mathematics Learning Centre, University of Sydney1

1 Exponents

1.1 Introduction

Whenever we use expressions like 7

3 or 2 5 we are using exponents.

The symbol 2

5 means 2×2×2×2×2

5 factors

. This symbol is spoken as 'two raised to the power five", 'two to the power five" or simply 'two to the five". The expression 2 5 is just a shorthand way of writing 'multiply 2 by itself 5 times". The number 2 is called thebase, and 5 theexponent.

Similarly, ifbis any real number thenb

3 stands forb×b×b. Herebis the base, and 3 the exponent.

Ifnis a whole number,b

n stands forb×b×···×b nfactors . We say thatb n is written in exponential form, and we callbthe base andnthe exponent, power or index. Special names are used when the exponent is 2 or 3. The expressionb 2 is usually spoken as 'bsquared", and the expressionb 3 as 'bcubed". Thus 'two cubed" means 2 3 =2×2×2=8.

1.2 Exponents with the Same Base

We will begin with a very simple definition. Ifbis any real number andnis a positive integer thenb n meansbmultiplied by itselfntimes. The rules for the behaviour of exponents follow naturally from this definition. First, let"s try multiplying two numbers in exponential form. For example 2 3 ×2 4 =(2×2×2)×(2×2×2×2) =2×2×2×2×2×2×2

7 factors

=2 7 =2 3+4 Examples like this suggest the following general rule.

Rule 1:b

n ×b m =b n+m That is, tomultiplytwo numbers in exponential form (with the same base), weaddtheir exponents. Let"s look at what happens when we divide two numbers in exponential form. For example, 3 6 3 4 =3×3×3×3×3×3

3×3×3×3

=3×3×3×3×3×3

3×3×3×3

=3×3 =3 2 =3 6-4 Mathematics Learning Centre, University of Sydney2

This leads us to another general rule.

Rule 2:

b n b m =b n-m In words, todividetwo numbers in exponential form (with the same base) , wesubtract their exponents. We have not yet given any meaning to negative exponents, sonmust be greater thanm for this rule to make sense. In a moment we will see what happens ifnis not greater than m. Now look at what happens when a number in exponential form is raised to some power.

For example,

(2 2 3 =(2×2)×(2×2)×(2×2) =2 6 =2

2×3

This suggest another general rule.

Rule 3:(b

m n =b mn That is, to raise a number in exponential form to a power, wemultiplythe exponents.

Examples

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