Logarithms - University of Plymouth
Levelling-Up
Basic Mathematics
Logarithms
Robin Horan
The aim of this document is to provide a short, self assessment programme for students who wish to acquire a basic competence in the use of logarithms.
Copyright c 2000 rhoran@plymouth.ac.uk Last Revision Date: January 16, 2001
Version 1.00
Table of Contents
1. Logarithms 2. Rules of Logarithms 3. Logarithm of a Product 4. Logarithm of a Quotient 5. Logarithm of a Power 6. Use of the Rules of Logarithms 7. Quiz on Logarithms 8. Change of Bases
Solutions to Quizzes Solutions to Problems
Section 1: Logarithms
3
1. Logarithms (Introduction)
Let a and N be positive real numbers and let N = an. Then n is called the logarithm of N to the base a. We write this as
Examples 1
n = loga N.
(a) Since 16 = 24, then 4 = log2 16.
(b) Since 81 = 34, then 4 = log3 81.
(c)
Since
3
=
9
=
91 2
,
then
1/2
=
log9
3.
(d) Since 3-1 = 1/3, then -1 = log3(1/3).
Section 1: Logarithms
4
Exercise
Use the definition of logarithm given on the previous page to determine the value of x in each of the following.
1. x = log3 27 2. x = log5 125 3. x = log2(1/4) 4. 2 = logx(16) 5. 3 = log2 x
Section 2: Rules of Logarithms
5
2. Rules of Logarithms
Let a, M, N be positive real numbers and k be any number. Then the following important rules apply to logarithms.
1. loga M N = loga M + loga N
2.
loga
M N
= loga M - loga N
3. loga mk = k loga M
4.
loga a = 1
5.
loga 1 = 0
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