Definition of a Logarithmic FunctionFor x > 0 and b > 0, b≠ 1,

y = logb x is equivalent to by = x

The function f (x) = logb x is the logarithmic function with base b.

Date:3.3: Logarithmic Functions and Their Graphs (3.3)

Logarithmic: y = logb x Exponential: by = x.Form Form

Logarithmic: y = logb x Exponential: by = x.Form Form

Exponent Exponent

Base Base

Text ExampleWrite each equation in its equivalent exponential form.a. 3 = log7 x b. 2 = logb 25 c. log4 26 = y

y = logb x means by = x

or y = log4 2673 = x b2 = 254y = 26

Write each equation in its equivalent logarithmic form.a. 122 = x b. b3 = 8 c. ey = 9

2 = log12 x 3 = logb 8 y = loge 9

Evaluatea. log2 16 b. log3 9 c. log25 5

Solution

log25 5 = 1/2 because

251/2 = 5.

25 to what power is 5?

log3 9 = 2 because 32 = 9.3 to what power is 9?

log2 16 = 4 because 24 = 16.2 to what power is 16?

c. log25 5

b. log3 9

a. log2 16

Logarithmic Expression Evaluated

Question Needed for Evaluation

Logarithmic Expression

Text Example

= 4 = 2 = 1/2

Basic Logarithmic Properties Involving One

logb b = 1because 1 is the exponent to which b must be

raised to obtain b (b1 = b)

logb 1 = 0because 0 is the exponent to which b must be

raised to obtain 1 (b0 = 1)

Inverse Properties of LogarithmsFor x > 0 and b 1,

logb by = yThe logarithm with base b of b raised to a power

equals that power.

b logb x = xb raised to the logarithm with base b of a number

equals that number.

Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = logbx

• The x-intercept is 1. There is no y-intercept.• The y-axis is a vertical asymptote.• If b > 1, the function is increasing. If 0 < b < 1, the

function is decreasing.• The graph is smooth and continuous. It has no

sharp corners or edges.

-2 -1

6

2 3 4 5

5

4

3

2

-1

-2

6

f (x) = 2x

g (x) = log2 x

y = x

-2 -1

6

2 3 4 5

5

4

3

2

-1

-2

6

f (x) = 2x

g (x) = log2 x

y = x

Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.

Solution We first set up a table of coordinates for f (x) = 2x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log2 x.

4

2

8211/21/4f (x) = 2x

310-1-2x

2

4

310-1-2g(x) = log2 x

8211/21/4x

Reverse coordinates.

Text Example

Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.

The graph of the inverse can also be drawn by reflecting the graph of f(x) = 2x over the line y = x.

Complete Student CheckpointWrite each equation in its equivalent logarithmic form.a. 25 = x b. b3 = 27 c. ey = 33

5 = log2 x 3 = logb 27 y = loge 33

Evaluatea. log10 100 b. log3 3 c. log36 6= 2 = 1 = 1/2

10? = 100 3? = 3 36? = 6

Complete Student CheckpointGraph f (x) 3x and on the same graph h(x) log

3x

x y

-2 1/9

-1 1/30 1

1 32 9

f(x) For h(x) just interchangethe x and y values of f(x)

f(x)

-21/9

-11/301

1329

x y

h(x)

h(x)

What is the Domain for the following functions?(remember you can’t take the log of a negative number)

Finding Domains

f(x) = log2x

g(x) = log2(x-1)

h(x) = log2(-x)

j(x) = - log2x

p(x) = log2x + 1

r(x) = 2log2x

x > 0

x > 1

x < 0

x > 0

x > 0

x > 0

g(x) = log4(x+3)

g(x) = log7(x-2)

g(x) = ln(3-x)

g(x) = ln(x-2)2

x > -3

x > 2

x < 3

all real numbers

Complete Student CheckpointFind the domain of h(x) log

4(x 5)

x - 5 >0 x > 5

Transformations Involving Logarithmic Functions

• Shifts the graph of f (x) = logbx upward c units if c > 0.

• Shifts the graph of f (x) = logbx downward c units if c < 0.

g(x) = logbx+cVertical translation

• Reflects the graph of f (x) = logbx about the x-axis.

• Reflects the graph of f (x) = logbx about the y-axis.

g(x) = - logbx

g(x) = logb(-x)

Reflecting

Multiply y-coordintates of f (x) = logbx by c,

• Stretches the graph of f (x) = logbx if c > 1.

• Shrinks the graph of f (x) = logbx if 0 < c < 1.

g(x) = c logbxVertical stretching or shrinking

• Shifts the graph of f (x) = logbx to the

left c units if c > 0, asymptote x = -c

• Shifts the graph of f (x) = logbx to the

right c units if c < 0, asymptote x = c

g(x) = logb(x+c)Horizontal translation

DescriptionEquationTransformation

Graph f (x) = log2 x and g(x) = 1 + -2log2(x - 1) in the same rectangular coordinate system.

Text Example

2

1

0

-1

4

2

1

1/2

y x

f(x)

f(x)

g(x)

What are the transformations for g(x)?Right 1Stretch x2

Reflect over x-axisUp 1

Properties of Common Logarithms(base is 10, log10 )

General Properties Common Logarithms

1. logb 1 = 0 1. log 1 = 0

2. logb b = 1 2. log 10 = 1

3. logb bx = x 3. log 10x = x4. b logb x = x 4. 10 log x = x

The Log on your calculator is base 10

Examples of Logarithmic Properties

log b b = 1

log b 1 = 0

log 4 4 = 1

log 8 1 = 0

3 log 3 6 = 6

log 5 5 3 = 3

2 log 2 7 = 7

Properties of Natural Logarithms

ln = loge General Properties Natural Logarithms

1. logb 1 = 0 1. ln 1 = 0

2. logb b = 1 2. ln e = 1

3. logb bx = x 3. ln ex = x4. b logb x = x 4. e ln x = x

Examples of Natural Logarithmic Properties

ln e =

ln 1 = 0

e ln 6 = 6

ln e 3 = 3

loge e1

log e 1

e log e 6

log e e 3

Complete Student CheckpointUse inverse properties to solve:

ln e25x eln x

25x x

Logarithmic Functions and Their Graphs