Cambridge Additional Mathematics

y

1 x

1

(e 1),

(1 e),

y=x

y=ex

y=lnx

O

142 Logarithms (Chapter 5)

InChapter 4we came across thenatural exponential e¼ 2 :718 28.

Given the exponential function f(x)=ex, the inverse function f¡^1 = logex is the logarithm in basee.

We use lnx to represent logex, and calllnxthenatural logarithmofx.

y=lnx is the reflection of y=ex in the mirror line y=x.

Notice that: ² ln 1 = lne^0 =0

² lne=lne^1 =1

² lne^2 =2

² ln

p

e=lne

1

(^2) =^1
2
² ln
³ 1
e
́
=lne¡^1 =¡ 1
lnex=x and elnx=x.
Since ax=
¡
elna
¢x
=exlna, ax=exlna, a> 0.

EXERCISE 5E.1

1 Without using a calculator find:

a lne^2 b lne^3 c ln

p

e d ln 1

e ln

³ 1

e

́

f ln^3

p

e g ln

³ 1

e^2

́

h ln

μ

1

p

e

¶

Check your answers using a calculator.

2 Simplify:

a eln 3 b e2ln3 c e¡ln 5 d e¡2ln2

3 Explain why ln(¡2) and ln 0 cannot be found.

4 Simplify:

a lnea b ln(e£ea) c ln

¡

ea£eb

¢

d ln(ea)b e ln

μ

ea

eb

¶

Example 14 Self Tutor

Use your calculator to write the following in the formek wherekis correct to

4 decimal places:

a 50 b 0 : 005

a 50

=eln 50 fusing x=elnxg

¼e^3 :^9120

b 0 : 005

=eln 0:^005

¼e¡^5 :^2983

E Natural logarithms

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_05\142CamAdd_05.cdr Friday, 20 December 2013 1:05:03 PM BRIAN