Cambridge International Mathematics

Thelogarithmic functionis f(x) = logaxwhere a> 0 , a 6 =1.

Consider f(x) = log 2 x which has graph y= log 2 x.

Since y= log 2 x,x=2y, we can obtain the table of values:

y ¡ 3 ¡ 2 ¡ 1 0 1 2 3

x^1814121248

Notice that:

² the graph of y= log 2 x is asymptotic to they-axis

² the domain of y= log 2 x is fxjx> 0 g

² the range of y= log 2 x is fyjy 2 Rg

THE INVERSE FUNCTION OF f(x) = logax

Given the functiony= logax, the inverse is x= logay finterchangingxandyg

) y=ax

So, f(x) = logax,f¡^1 (x)=ax

B THE LOGARITHMIC FUNCTION [3.10]

x

2 4 6 8

4

2

-2

-4

y

O

yx¡=¡logx

Logarithms (Chapter 31) 627

m log 7

¡p 3

7

¢

n log 2 (4

p

2) o logp 22 p log 2

³

1

4

p

2

́

q log 10 (0:01) r logp 24 s logp 3

¡ 1

3

¢

t log 3

³

1

9

p

3

́

4 Rewrite as logarithmic equations:

a y=4x b y=9x c y=ax d y=(

p

3)x

e y=2x+1 f y=3^2 n g y=2¡x h y=2£ 3 a

5 Rewrite as exponential equations:

a y= log 2 x b y= log 3 x c y= logax d y= logbn

e y= logmb f T= log 5

¡a

2

¢

g M=^12 log 3 p h G= 5 logbm

i P= logpbn

6 Rewrite the following, makingxthe subject:

a y= log 7 x b y=3x c y=(0:5)x d z=5x

e t= log 2 x f y=2^3 x g y=5

x 2

h w= log 3 (2x)

i z=^12 £ 3 x j y=^15 £ 4 x k D= 101 £ 2 ¡x l G=3x+1

7 Explain why, for all a> 0 , a 6 =1: a loga1=0 b logaa=1

IGCSE01

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Y:\HAESE\IGCSE01\IG01_31\627IGCSE01_31.CDR Tuesday, 18 November 2008 11:12:17 AM PETER