Cambridge Additional Mathematics

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means

138 Logarithms (Chapter 5)

6 If x= log 2 P, y= log 2 Q, and z= log 2 R, write in terms ofx,y, andz:

a log 2 (PR) b log 2 (RQ^2 ) c log 2

μ

PR

Q

¶

d log 2 (P^2

p

Q) e log 2

μ

Q^3

p

R

¶

f log 2

μ

R^2

p

Q

P^3

¶

7 If p= logb 2 , q= logb 3 , and r= logb 5 , write in terms ofp,q, andr:

a logb 6 b logb 45 c logb 108

d logb

³

5

p

3

2

́

e logb

¡ 5

32

¢

f logb(0:2)

8 If logtM=1: 29 and logtN^2 =1: 72 , find:

a logtN b logt(MN) c logt

μ

N^2

p

M

¶

9 Suppose logbP=5 and logb(P^3 Q^2 )=21. Find logbQ.

10 Suppose that logt(AB^3 )=15 and logt

μ

A^2

B

¶

=9.

a Writetwoequations connecting logtA and logtB.

b Find the values of logtA and logtB.

c Find logt

¡

B^5

p

A

¢

d WriteBin terms oft.

We can use the laws of logarithms to write equations in a different form. This can be particularly useful if

an unknown appears as an exponent.

For the logarithmic function, for every value ofy, there is only one corresponding value ofx. We can

therefore take the logarithm of both sides of an equation without changing the solution. However, we can

only do this if both sides are positive.

Example 10 Self Tutor

Write these as logarithmic equations (in base 10 ):

a y=5£ 3 x b P=p^20

n

a y=5£ 3 x

) lgy=lg(5£ 3 x)

) lgy=lg5+lg3x

) lgy=lg5+xlg 3

b P=p^20

n

) lgP=lg

μ 20

n

1

2

¶

) lgP=lg20¡lgn

1

2

) lgP=lg20¡^12 lgn

D Logarithmic equations

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_05\138CamAdd_05.cdr Tuesday, 21 January 2014 2:47:49 PM BRIAN