In the lessons covering Exponential Functions (Lessons 29 ...

16-week Lesson 31 (8-week Lesson 25)

Logarithmic Functions

In the lessons covering Exponential Functions (Lessons 29 & 30), we

determined that exponential functions are one-to-one. And in Lesson 27,

we determined that if a function is one-to-one (like exponential functions

are), it must have an inverse. Therefore exponential functions will have an

inverse.

To find the inverse of an exponential function, we can follow the same

steps that we have used before to find the inverse of a function (change the

function to an equation, solve for ?, and then change back to a function).

Example 1: List the domain and range of the function ? (? ) = 2? . Then

find its inverse function ? ?1 (? ) and list its domain and range.

?(?) = ??

In order to find the inverse of an exponential function, we will

once again write the function as an equation and solve for ?.

?(? ) = 2?

? = 2?

Unfortunately we can��t use any of our current techniques to

bring the exponent ? down in order to isolate it. Instead we

need to use a new technique which is to convert the

exponential equation ? = 2? to logarithmic form, since a

logarithmic function is the inverse of an exponential function.

?(? ) = 2?

? = 2?

��

? = log 2 (?)

Converting our exponential equation to a logarithmic equation

allowed us to isolate the variable ?. This also shows that our

logarithm (log 2 (?)) is equal to ?, which used to be our

exponent. So a logarithm is simply an exponent.

?(?) = ??

??? (?) =

Domain of ?(?) = ?? :

Domain of ??? :

(the domain is the set of

inputs, and the inputs of this

function are exponents)

(the domain is the set of inputs,

and the inputs of this function

are powers of the base 2)

Range of ?(?) = ?? :

Range of ??? :

(the range is the set of outputs,

and the outputs of this function

are powers of the base 2)

(the range is the set of

outputs, and the outputs of this

function are exponents)

1

16-week Lesson 31 (8-week Lesson 25)

Logarithmic Functions

For any exponential function ?(? ) = ? ? , where ? is a positive number

other than 1, its inverse function will be ? ?1 (? ) = log ? (? ).

?(?) = ??

??? (?) = ??? ? (?)

Domain of ?(?) = ?? :

Domain of ??? :

a.

(?��, ��)

(?, ��)

Range of ?(?) = ?? :

b.

Range of ??? :

(?, ��)

(?��, ��)

Logarithmic Functions:

- the inverse of an exponential function

o the function that undoes an exponential function

? if ?(? ) = ? ? , then ? ?1 (? ) = log ? (? )

- a logarithm is simply an exponent; specifically, log ? (? ) represents

the exponent that makes the base ? equal to the argument ?

- the quantity in parentheses (? in this case) is called the argument, and

the argument is a power of the base ?

o just like the base of an exponential function, the base of a

logarithm can be any positive number other than 1, and since the

argument is a power of that base, THE ARGUMENT MUST

BE POSITIVE

Exponential Functions

Logarithmic Functions

Any real number can be the input

of an exponential function

?(? ) = ? ? , but only positive

outputs are possible because the

base ? must be a positive number,

and a positive number taken to the

power of any real number is

ALWAYS positive.

Only positive real numbers can be

the argument of a logarithmic

function ?(? ) = log ? (? ) because

the argument is a power of the

positive base ?, but any real

number is possible for the output

because the output of a logarithmic

function is the same as the input of

its inverse (an exponential

function), an exponent.

2

16-week Lesson 31 (8-week Lesson 25)

Logarithmic Functions

Example 2: Given the logarithmic function ? (? ) = log 36 (? ), find the

following function values. If no function value exists for a particular

input, write DNE.

a. ? (36) = log 36 (36)

A logarithm is an exponent. In this case, what exponent makes the

base of 36 equal to the argument of 36? In other words, 36? = 36

??? ?? (??) = ?, because ? is the exponent that make ?? equal to ??

1

1

b. ? (36) = log 36 (36)

Again, a logarithm is an exponent. In this case, what exponent makes

1

1

the base of 36 equal to the argument of 36? In other words, 36? = 36.

In part a. we found that 361 = 36, so 1 is the exponent that makes 36

equal to itself, but what exponent makes 36 equal to it��s reciprocal?

c. ? (?36) = log 36 (?36)

Again, a logarithm is an exponent. In this case, what exponent makes

the base of 36 equal to the argument of ?36? In other words,

36? = ?36 .

Is there an exponent that will make a positive base like 36 become

negative?

d. ? (0) = log 36 (0)

A logarithm is an exponent. What exponent makes the base of 36

equal to the argument of 0 (36? = 0)? Is there an exponent that will

change a positive base like 36 equal to 0?

e. ? (6) = log 36 (6)

What exponent makes 36 equal to 6 (36? = 6)? 6 is the square root

of 36, so what exponent is the equivalent of a square root?

3

16-week Lesson 31 (8-week Lesson 25)

Logarithmic Functions

?(? ) = log ? (? ) represents the exponent that makes the base ? equal to the

argument ?. Using the logarithmic function from Example 3,

?(? ) = log 36 (? ) represents the exponent that makes the base 36 equal to

the argument ?:

?(1296) = log 36 (1296), which equals 2 because 362 = 1296

1

1

1

? (6) = log 36 (6), which equals ? 2 because 36

?

1

2

1

=6

So the logarithmic function ? (? ) = log 36 (? ) is basically saying 36 ? = ?,

which means that ? must ALWAYS be a positive value, because 36 to the

power of any exponent will ALWAYS produce a positive value. This will

be true for any logarithmic function ?(? ) = log ? (? ).

?(? ) = log ? (? )

? ?1 (? ) = ? ?

Domain: (0, ��)

Domain: (?��, ��)

Range: (?��, ��)

Range: (0, ��)

? is a power of the base ?, and

since ? is positive, ? must be

positive. The output ?(? ) is the

exponent which produces that

power, and an exponent can be any

real number, since exponents are

never restricted.

? is an exponent of the base ?, so ?

is unrestricted. The output ?(? ) is

a power of ?, and since ? is

positive, ? ? must be positive as

well.

Understanding that the argument of a logarithmic function is ALWAYS

positive will become even more important when we start solving

logarithmic equations. At that point, we will always have to check our

answers to verify they result in a positive argument.

4

16-week Lesson 31 (8-week Lesson 25)

Logarithmic Functions

Example 3: Given the logarithmic function ? (? ) = log100 (? ), find the

following function values. If no function value exists for a particular

input, write DNE.

1

1

a. ? (100) = log100 (100)

b. ? (100) = log100 (100)

b. 1

1

100? = 100

100? = 100

1

c. ? (?100) = log100 (?100)

1

d. ? (? 100) = log100 (? 100)

d.

1

100? = ?100

100? = ? 100

e. ? (1) = log100 (1)

f. ? (0) = log100 (0)

f.

100? = 1

100? = 0

g. ? (10,000) = log100 (10,000)

h. ? (

1

1

) = log100 (10,000)

10,000

h.

1

100? = 10,000

100? = 10,000

1

i. ? (10) = log100 (10)

1

j. ? (10) = log100 (10)

j.

1

100? = 10

100? = 10

1

k. ? (?10) = log100 (?10)

1

l. ? (? 10) = log100 (? 10)

l. 1

1

100? = ?10

100? = ? 10

A logarithm is an exponent; a logarithm represents the exponent needed to

change a base into a power of that base (the argument). When finding

function values for a logarithmic function, think about what exponent is

needed to make the base equal to the argument.

5

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