Cambridge Additional Mathematics

Logarithms (Chapter 5) 131

Numbers like

p

10 , 10

p

10 and

1

p 510 can also be written in the form 10

x as follows:

p

10 = 10

1

2

=10^0 :^5

10

p

10 = 10^1 £ 100 :^5

=10^1 :^5

1

p 510 =10

¡^15

=10¡^0 :^2

In fact, all positive numbers can be written in the form 10 x.

Thelogarithm in base 10 of a positive number is the power that 10 must be raised to in order to

obtain the number.

For example:

² Since 1000 = 10^3 , we write log 10 (1000) = 3

or lg(1000) = 3:

² Since 0 :01 = 10¡^2 , we write log 10 (0:01) =¡ 2

or lg(0:01) =¡ 2 :

We hence conclude that:

lg 10x=x for any x 2 R.

a=10lga for any a> 0.

Example 1 Self Tutor

Without using a calculator, find:

a log 100 b log(^4

p

10)

a log 100 = log 10^2 =2 b log(^4

p

10) = log(10

1

(^4) )=^1
4

EXERCISE 5A

1 Without using a calculator, find:

a lg 10 000 b lg 0: 001 c lg 10 d lg 1

e lg

p

10 f lg(^3

p

10) g lg

μ

1

p (^410)
¶
h lg
¡
10
p
10
¢
i lg^3
p
100 j lg
μ
100
p
10
¶
k lg
¡
10 £^3
p
10
¢
l lg
¡
1000
p
10
¢
Check your answers using your calculator.
2 Simplify:
a lg 10n b lg (10a£100) c lg
³
10
10 m
́
d lg
μ
10 a
10 b
¶
lg log
10 0
means.
must be positive since
for all.
aa
a

x
qp
x 2 R
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Y:\HAESE\CAM4037\CamAdd_05\131CamAdd_05.cdr Tuesday, 21 January 2014 2:47:20 PM BRIAN