Section 5.4Logarithmic Functions

3 21a log 81 b log8

(a) 3 raised to what power yields 81? 3log 813 81yy

43 3

4

y

y 3Therefore, log 81 4

32 23

y

y

2

1Therefore, log 38

(b) 2 raised to what power yields ?18

2

33

1log8

1 12 28 2

y

y

3 2

2

3a log 2 b log1

c log 1

xf x x F xx

h x x

a The domain of consists of all for which 2 0.f x x

2 or 2,x

3b The domain of is restricted to 01

xFx

, 3 1, .

c Since the absolute value function is never negative, the domain would consist of all real numbers except 1 0.x

,1 1,

Natural Logarithm Function

2 3 so 5.x x

The domain of is 2, .f

The range of f is all real numbers and the vertical asymptote is x = 2.

ln 2f x x

The inverse implicitly is ln 2x y

2xe y

ln 2x y

12xe y f x

Since the range of f equals the domain of f -1, the range of f is all real numbers.

1 2xf x e

Common Logarithm Function

2Solve: log 2 1 3 log 343 3xa x b

2(a) Change log 2 1 3 to exponential form.x

32 2 1x 8 2 1x 72

x

2 27log 2 1 log 8 32

(b) Change log 343 3 to exponential form.x

3 343x 7x

7log 343 3

3Solve: 2 6xe

3 3xe Isolate the exponential.

ln 3 3x Change to logarithmic form.

ln 33

x Exact solution

0.366 Approximate solution