Section 4.6 Logarithmic Functions

Section 4.6

Logarithmic Functions

Section 4.6

Logarithmic Functions

Objectives:1. To convert between exponential

and logarithmic form.2. To evaluate logarithms of

numbers.3. To graph logarithmic functions

and give their domains and ranges.

Objectives:1. To convert between exponential

and logarithmic form.2. To evaluate logarithms of

numbers.3. To graph logarithmic functions

and give their domains and ranges.

The inverse of the exponential function is the logarithmic function.

The inverse of the exponential function is the logarithmic function.

y = x

y = xf(x) = 2xf(x) = 2x

f(x) = log2xf(x) = log2x

Logarithmic function f(x) = logax, where a > 0, a 1, and x > 0. It is the inverse of the exponential function f(x) = ax.

Logarithmic function f(x) = logax, where a > 0, a 1, and x > 0. It is the inverse of the exponential function f(x) = ax.

DefinitionDefinitionDefinitionDefinition

The rule y = logax is equivalent to ay = x; therefore, a is the base of the logarithm and y, the logarithm, is an exponent.

The rule y = logax is equivalent to ay = x; therefore, a is the base of the logarithm and y, the logarithm, is an exponent.

The logarithmic expression can be written in either of two forms.

The logarithmic expression can be written in either of two forms.

Log Form Exponential Form

logax = y ↔ay = x

Log Form Exponential Form

logax = y ↔ay = x

EXAMPLE 1 Write log4 64 = y in exponential form.EXAMPLE 1 Write log4 64 = y in exponential form.

log4 64 = y

4y = 64

log4 64 = y

4y = 64

3-5 =3-5 =1

2431

243

log3 = -5log3 = -51

2431

243

EXAMPLE 2 Write 3-5 = in logarithmic form.EXAMPLE 2 Write 3-5 = in logarithmic form.

12431

243

EXAMPLE 3 Prove loga 1 = 0 a 0 with a 1.EXAMPLE 3 Prove loga 1 = 0 a 0 with a 1.

a0 = 1

loga 1 = 0

a0 = 1

loga 1 = 0

A logarithm in base ten is called a common logarithm. The base of a common log is not usually written. Common logs can be found using the log key on your calculator.

A logarithm in base ten is called a common logarithm. The base of a common log is not usually written. Common logs can be found using the log key on your calculator.

The second special type of logarithm is the natural logarithm whose base is e. Recall that e is an irrational number that is approximately 2.71828. The special notation for a natural logarithm is ln.

The second special type of logarithm is the natural logarithm whose base is e. Recall that e is an irrational number that is approximately 2.71828. The special notation for a natural logarithm is ln.

e, also called the Euler number, is defined as e, also called the Euler number, is defined as

oror

nn

nn nn11

1 +1 +limlime =e =

∙∙∙∙∙∙++++++242411

6611

2211

= 1 + 1 += 1 + 1 +n!n!11

e =e =n=0n=0

Homework

pp. 202-203

Homework

pp. 202-203

►A. ExercisesChange the following logarithms to exponential form.

1. log2x = y


1. log2x = y


3. log4 = -2


3. log4 = -2 1

161

16


5. log 1000 = 3


5. log 1000 = 3


7. ln 1 = 0


7. ln 1 = 0

►A. ExercisesExplain why the following are true.

9. loga a = 1

►A. ExercisesExplain why the following are true.

9. loga a = 1

►A. ExercisesChange the following exponential expressions to log form.11. 53 = 125






►B. ExercisesGraph. Give the domain and range of each.17. y = log4x




►B. ExercisesEvaluate.21. log5 1,953,125

►B. ExercisesEvaluate.21. log5 1,953,125

5y = 1,953,125

5y = 59

y = 9

5y = 1,953,125

5y = 59

y = 9

2y = 8-1

2y = (23)-1

2y = 2-3

y = -3

2y = 8-1

2y = (23)-1

2y = 2-3

y = -3

►B. ExercisesEvaluate.23. log2


1818



11296

11296

■ Cumulative ReviewUse interval notation to show the intervals of continuity of the following functions.

29. f(x) =

■ Cumulative ReviewUse interval notation to show the intervals of continuity of the following functions.

29. f(x) = 3x2 – 5x

3x2 – 5x

■ Cumulative ReviewUse interval notation to show the intervals of continuity of the following functions.30. g(x) = 3x2 – 7x + 9

■ Cumulative ReviewUse interval notation to show the intervals of continuity of the following functions.30. g(x) = 3x2 – 7x + 9

■ Cumulative ReviewUse interval notation to show the intervals of continuity of the following functions.31. h(x) = x – 3

■ Cumulative ReviewUse interval notation to show the intervals of continuity of the following functions.31. h(x) = x – 3

■ Cumulative Review32. Name three characteristics of a graph

that cause a function to be discontinuous. Name an example of each.

■ Cumulative Review32. Name three characteristics of a graph

that cause a function to be discontinuous. Name an example of each.

■ Cumulative Review33. Certain functions, such as the

absolute value function, do not graph as a smooth curve, but have sharp turns in their graphs. Is such a function discontinuous?

■ Cumulative Review33. Certain functions, such as the

absolute value function, do not graph as a smooth curve, but have sharp turns in their graphs. Is such a function discontinuous?

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Section 4.6 Logarithmic Functions