10.8 THE NATURAL LOG FUNCTION

e is : irrational #

like , not variable

2.718

NATURAL LOG

Recall common log:log 2 = log10 2

Natural log: denotedloge x ln x

ln e

BASE E

If ln x = 5, then loge x = 5

then ln 7 = xIf e x = 7,

= lne e= 1

ln ek= k(ln e)

= k(1) = k

e ln 7 = e lne 7 = 7

Same laws & properties of logs apply

to natural logsthen (around the world!), so e 5 = x

then loge 7 = x

EXAMPLE 1) SIMPLIFY

21lne

2lne e

2 lne e 2(1) 2

2ln5 ln 4 3

EX 2) WRITE AS A SINGLE LOG

2 3ln 5 ln 4 lne e e e

33 lne e

3

100ln

e

3

25 4lne e

3

100lne e

Write as ln

( )-12 1e x

ln 2x

SOLVE—LEAVE ANSWERS IN TERMS OF “E”

ln 2e x

2e x

1ln 2x Ex 3) Ex 4)

1ln 2e x

2 1xe

( )-1

2e x

2 ln ln 9e ex e

EX 4) SOLVE (WRITE IN TERMS OF LN)

2ln ln 9xe ee

2 9xe

ln 3x

2 ln 9ex 12 ln 9ex

12ln 9ex

ln 3ex

Take natural log of both sides

ln 1e e

129 9

ln 5x

TOO

5e x

3 64xe 1) 2)

ln 4 x

1ln 7 ln 2 ln8

3

TOO—WRITE AS A SINGLE LOG

7 2ln

2e

ln 7

HOMEWORK

#6 Pg. 490 1 – 33 odd