Lesson #1: Exponential Functions and Their
Inverses Day 1
Unit 5:
Logarithmic Functions
Lesson #1: Exponential Functions and Their Inverses
Day 1
Exponential Functions & Their Inverses


Exponential Functions are in the form
.
The inverse of an exponential is a reflection in the line
. The and coordinates are swapped.
Recall, the domain and range
are also swapped.
Inverse of the
exponential function
:
How do we solve for ?
Algebra II with Trigonometry
Unit 8
1
Lesson #1: Exponential Functions and Their
Inverses Day 1

To describe
2 in words:

is the exponent to the base 2 needed to obtain .

We can re-write this sentence using the word logarithm,
since logarithm means exponent.

is the logarithm to the base 2 of .

To abbreviate and write in symbolic form:

is the logarithm to the base of .
log
Inverse Function of the Exponential Function log
These two equations are equivalent
1).
if
0 and
0, (
Remember: A logarithm IS an exponent.
: exponent
: base
: answer
Algebra II with Trigonometry
Unit 8
2
Lesson #1: Exponential Functions and Their
Inverses Day 1
Ex 1: Write the equation of 1. Switch and . of
6
of log
of 4 . of log
.
6
2. Solve for . log
Ex 2: Write the equation of 1. Switch and . .
log
2. Solve for . 9
Ex 3: Write the equation of log
Ex 4: Write the equation of .
1
2
Algebra II with Trigonometry
Unit 8
3
Lesson #1: Exponential Functions and Their
Inverses Day 1
Ex 5: Solve each equation for in terms of . Important!
10
(a) These equations
are equivalent.
log
Note: A logarithm of log
base 10 is called a common logarithm.
4
(b) 1
4
⇒
log
Ex 6: Solve each equation for in terms of (a) log
Important!
5
(b) .
log
These equations are equivalent.
.
0.1
Algebra II with Trigonometry
Unit 8
4
Lesson #1: Exponential Functions and Their
Inverses Day 1
Recall:
log
These two equations are equivalent
1).
if
0 and
0, (
Remember: A logarithm IS an exponent.
: exponent
: base
Important!
You must be able to go back and : answer
forth between equivalent logarithmic & exponential forms! Ex 1: Write each logarithmic equation in the equivalent exponential form. (a) log 1
8
(b) (c) log
1
log 81
3
Algebra II with Trigonometry
Unit 8
0
81
1
16
2
4
(d) log
6
36
4
1
16
1
2
6
5
Lesson #1: Exponential Functions and Their
Inverses Day 1
Ex 2: Write each exponential equation in the equivalent logarithmic form. (a) 5
625
log 625
(b) 10
log
1
49
(c) 7
4
1
49
log
1,000
1,000
3
log 1,000
3
(d) 9
(a) log 64
(b) log
1. Set expression equal to a
variable or .
log 64
6
2. Re-write logarithmic equation
in equivalent exponential form.
2
64
3. Solve exponential equation.
(Recall:Needcommonbases!)
2
2
6
Algebra II with Trigonometry
Unit 8
3
log 3
Ex 3: Evaluate each expression. log
2
1
2
1
8
1
8
3
2
1
8
2
2
3
6
Lesson #1: Exponential Functions and Their
Inverses Day 1
Ex 4: Evaluate each expression. (a) log 27
(b) 2 log 81
log 27
log 81
1
3
27
3
81
3
3
3
3
4
3
2 4
3
8
Ex 5: Solve each equation for . 5
2
(a) log
(b) log 9
1. Re-write logarithmic equation
in equivalent exponential form.
1. Re-write logarithmic equation
in equivalent exponential form.
9
4
2. Solve the power equation.
2. Evaluate the expression.
(Recall:FractionalExponent
4
2
32
Algebra II with Trigonometry
Unit 8
2
)
(Recall:Usereciprocalexponent)
∙
9
*If we take
square root, we
must introduce .
*However, must consider restriction
on base of a logarithmic function!
3
7
Lesson #1: Exponential Functions and Their
Inverses Day 1
Ex 6: Solve each equation for . (a) log 3
8
(b) log
6
2
3
8
3
64
3
8
72
3
1
3
3
1
27
24
Ex 7: Evaluate the given expression:
log 27 8 log
log 256
2
⟹
3
8
4
1
4
5
4
Evaluate each expression individually.
Re-write each logarithmic expression
in equivalent exponential form.
log 27
3
27
3
3
3
Algebra II with Trigonometry
Unit 8
log 2
16
2
2
4
2
1
1
4
log 256
4
256
2
2
2
8
4
8