4.3 – LOGARITHMIC FUNCTIONSTarget Goals:

1. Use the relationship between exponential and logarithmic functions to change them into the opposite form

2. Evaluate logarithms with and without a calculator.

3. Solve logarithmic equations

EXPONENTIAL VS. LOGARITHMIC FORM

Exponential Form Logarithmic Form

ya x logay x

0, 1a a

EX 1) CHANGE EACH EXPONENTIAL EXPRESSION TO AN EQUIVALENT EXPRESSION INVOLVING A LOGARITHM:

64.5 x 9qe 5 14s

4.5log 6x log 9e q log 14 5s

ln 9 q

EX 2) CHANGE EACH LOGARITHMIC EXPRESSION TO AN EQUIVALENT EXPRESSION INVOLVING AN EXPONENT:

log 6 5x ln 2b 5log 4 x

5 6x log 2e b 5 4x

2e b

EX 3) FIND THE EXACT VALUE OF EACH:

3log 81 2

1log

32

3 81x

x x

43 3x 4x

12

32x

12 32x 152 2x

5x

DOMAIN AND RANGE OF LOG FUNCTIONS! Domain of a logarithmic function = range of the exponential function =

(0,∞)

Range of a logarithmic function = domain of the exponential function = (-∞, ∞)

Ex 4) Find the domain of each logarithmic function:3( ) log ( 8)f x x 5

2( ) log

3

xf x

x

8 0x

8x

(8, )

20

3

x

x

2 0x 2x

(2,3)

PROPERTIES OF A LOGARITHMIC FUNCTION:

1. The domain is the set of positive real numbers; the range is all real numbers.

2. The x-intercept of the graph is 1. There is no y-intercept.

3. The y-axis (x = 0) is a vertical asymptote of the graph.

4. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1.

5. The graph of f contains the points (1, 0), (a, 1) and (1/a, -1).

6. The graph is smooth and continuous, with no corners or gaps or holes.

( ) logaf x x

EX 5) SOLVE THE FOLLOWING EQUATIONS:3log (4 7) 2x log 64 2x 2 5xe

23 4 7x 9 4 7x 16 4x

4x

2 64x 8x 8x

ln 5 2x

ln 5

2x