Cambridge Additional Mathematics

If

then.

ln =

=

xa

xea

Logarithms (Chapter 5) 143

5 Use your calculator to write the following in the form ek wherekis correct to 4 decimal places:

a 6 b 60 c 6000 d 0 : 6 e 0 : 006

f 15 g 1500 h 1 : 5 i 0 : 15 j 0 :000 15

Example 15 Self Tutor

Findxif:

a lnx=2: 17 b lnx=¡ 0 : 384

a lnx=2: 17

) x=e^2 :^17

) x¼ 8 : 76

b lnx=¡ 0 : 384

) x=e¡^0 :^384

) x¼ 0 : 681

6 Findxif:

a lnx=3 b lnx=1 c lnx=0 d lnx=¡ 1

e lnx=¡ 5 f lnx¼ 0 : 835 g lnx¼ 2 : 145 h lnx¼¡ 3 : 2971

LAWS OF NATURAL LOGARITHMS

The laws for natural logarithms are the laws for logarithms written in basee:

For positiveAandB:

² lnA+lnB=ln(AB) ² lnA¡lnB=ln

μ

A

B

¶

² nlnA=ln(An)

Example 16 Self Tutor

Use the laws of logarithms to write the following as a single logarithm:

a ln 5 + ln 3 b ln 24¡ln 8 c ln 5¡ 1

a ln 5 + ln 3

=ln(5£3)

=ln15

b ln 24¡ln 8

=ln

¡ 24

8

¢

=ln3

c ln 5¡ 1

=ln5¡lne^1

=ln

¡ 5

e

¢

Example 17 Self Tutor

Use the laws of logarithms to simplify:

a 2ln7¡3ln2 b 2ln3+3

a 2ln7¡3ln2

=ln(7^2 )¡ln(2^3 )

=ln49¡ln 8

=ln

¡ 49

8

¢

b 2ln3+3

=ln(3^2 )+lne^3

=ln9+lne^3

=ln(9e^3 )

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_05\143CamAdd_05.cdr Tuesday, 21 January 2014 2:48:09 PM BRIAN