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Exponential and Logarithmic Functions

Properties of the
Logarithmic Function

•Properties of , a >1

•Domain: Positive real numbers ; (0, ∞)
•Range: All real numbers
•Intercepts:

•x- intercept of x = 1
•No y-intercepts

•y-axis is horizontal asymptote
•Increasing and one-to-one.
•Smooth and continuous
•Contains points (1,0), (a, 1) and

•Properties of , 0 <a <1

•Domain: Positive real numbers; (0, ∞)
•Range: All real numbers
•Intercepts:

•x-intercept of x= 1
•No y-intercepts

•y-axis is horizontal asymptote
•Decreasing and one-to-one.
•Smooth and continuous
•Contains points (1,0), (a, 1) and

Special Logarithm Functions

•Natural logarithm:

if and only if


Common logarithm :

if and only if

•Example. Graph the function

Problem:

Answer:

Logarithmic Equations

•Examples. Solve the logarithmic
equations. Give exact answers.

(a) Problem:

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:

•Examples. Solve the exponential
equations using logarithms . Give
exact answers.

(a) Problem:

Answer:

(b) Problem:

Answer:

Key Points

•Logarithmic Functions
•Domain and Range of Logarithmic Functions
•Graphing Logarithmic Functions
•Properties of the Logarithmic Function
•Special Logarithm Functions
•Logarithmic Equations

Properties of
Logarithms


Section 4.5

Properties of Logarithms

•Theorem. [Properties of Logarithms]
For a >0, a≠1, and r some real
number:


•Theorem. [Properties of Logarithms]
For M, N, a >0, a≠1, and r some
real number:

•Examples. Evaluate the following
expressions.

(a) Problem:

Answer:

(b) Problem:

Answer:

(c) Problem:

Answer:

•Examples. Evaluate the following
expressions if and
.

(a) Problem:

Answer:
(b) Problem:

Answer:

(c) Problem:

Answer:

•Example. Write the following
expression as a sum of logarithms .
Express all powers as factors .

Problem:

Answer:

•Example. Write the following
expression as a single logarithm.

Problem:

Answer:

•Theorem. [Properties of Logarithms]
For M, N, a >0, a≠1,

•If M= N, then
•If , then M= N

•Comes from fact that exponential and
logarithmic
functions are inverses.

Logarithms with Bases
Other than e and 10

•Example.
Problem: Approximate rounded to
four decimal places

Answer:

•Theorem. [ Change -of-Base Formula]
If a≠1, b≠1 and Mare all positive
real numbers, then


•In particular

•Examples. Approximate the following
logarithms to four decimal places

(a) Problem:

Answer:
(b) Problem:

Answer:

Key Points

•Properties of Logarithms
•Properties of Logarithms
•Logarithms with Bases Other than e
and 10

Logarithmic and
Exponential
Equations

Section 4.6

Solving Logarithmic Equations

•Example.
Problem: Solve
algebraically .

Answer:

•Example.

Problem: Solve
graphically.

Answer:

•Example.
Problem: Solve
algebraically.

Answer:

•Example.
Problem: Solve
graphically.

Answer:

Solving Exponential Equations

•Example.

Problem: Solve
algebraically.

Answer:

•Example.

Problem: Solve
graphically.

Answer:

•Example.

Problem: Solve algebraically. Give
an exact answer, then approximate your
answer to four decimal places.

Answer:

•Example.

Problem: Solve graphically.
Approximate your answer to four
decimal places.

Answer:

•Example.

Problem: Solve algebraically.
Give an exact answer, then approximate
your answer to four decimal places.

Answer:

•Example.

Problem: Solve graphically.
Approximate your answer to four
decimal places.

Answer:

•Example.

Problem: Solve algebraically.
Give an exact answer, then approximate
your answer to four decimal places.

Answer:

•Example.

Problem: Solve graphically.
Approximate your answer to four
decimal places.

Answer:

Key Points

•Solving Logarithmic Equations
•Solving Exponential Equations

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