Cambridge Additional Mathematics

Logarithms (Chapter 5) 145

5 Write the following equations without logarithms, assuming all terms are positive:

a lnD=lnx+1 b lnF=¡lnp+2 c lnP=2x+ln5

d lnM=2lny+3 e lnB=3t¡ln 4 f lnN=¡^13 lng

g lnQ¼3lnx+2: 159 h lnD¼ 0 :4lnn¡ 0 : 6582 i lnT¼¡x+1: 578

InChapter 4we found solutions to simple exponential equations where we could make equal bases and

then equate exponents. However, it is not always easy to make the bases the same. In these situations we

uselogarithmsto find the solution.

Example 20 Self Tutor

Solve forx, giving your answers correct to 3 significant figures:

a 2 x=7 b 53 x¡^1 =90

a 2 x=7

) lg 2x=lg7

) xlg 2 = lg 7 flg(an)=nlgag

) x=

lg 7

lg 2

) x¼ 2 : 81

b 53 x¡^1 =90

) lg 5^3 x¡^1 =lg90

) (3x¡1) lg 5 = lg 90 flg(an)=nlgag

) 3 x¡1=

lg 90

lg 5

) x=^13

³

1+

lg 90

lg 5

́

) x¼ 1 : 27

Example 21 Self Tutor

Findxexactly:

a ex=30 b 3 e

x

2

=21

a ex=30

) x=ln30

b 3 e

x

2

=21

) e

x

2

=7

)

x

2

=ln7

) x= 2 ln 7

EXERCISE 5F

1 Solve forx, giving your answer correct to 3 significant figures:

a 2 x=10 b 3 x=20 c 4 x= 100

d

¡ 1

2

¢x

=0: 0625 e

¡ 3

4

¢x

=0: 1 f 10 x=0:000 01

using logarithms F Solving exponential equations

LOGARITHMS

F

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_05\145CamAdd_05.cdr Tuesday, 21 January 2014 2:49:47 PM BRIAN