3-3 : Functions and their graphs

Lesson objectives

Students will be able to convert equations between logarithmic form and exponential form, evaluate common and natural logarithms, and graph common and natural

logarithmic functions.

Warm Up

Solve each equation.1. 8 = x3

2

2. x1/4 = 216

3. 27 = 3x 3

4. 46 = 43x

2

Quick Review

-2

11

32

0

3

4

Evaluate the expression without using a calculator.

1. 6

82.

23. 7

Rewrite as a base raised to a rational number exponent.

14.

5. 10

e

Slide 3- 4

Quick Review Solutions

3 / 2

1/

-2

11

3

4

2

0

3

4

Evaluate the expression without using a calculator.

1. 6

82.

23. 7

Rewrite as a base raised to a rational number exponent.

14.

5. 10

1

36

2

1

10

ee

Slide 3- 5

What you’ll learn about• Inverses of Exponential Functions• Common Logarithms – Base 10• Natural Logarithms – Base e• Graphs of Logarithmic Functions

… and whyLogarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.

Key Concepts :

Logarithm- has base b of a positive number y is defined as follows:

If = , then = x.

Common logarithm- a logarithm that uses base 10. ex. log 8

Logarithmic Functions are Inverses of Exponential Functions

If >0 and >0, ≠ 1, then = if and only if = . 𝑎 𝑏 𝑏 𝑦 𝑥 𝑥Graph:

𝑦 = 𝑥 𝑦 = 𝑥

Evaluating Logarithmic and Exponential Expressions

A. 8 = 3 because

B. = because

C. = -2 because

D. 7 = 1 because

E. = 11 because

Logarithmic functions are inverses of exponential functions (x & y are switched)

x-2-1012

Common Logarithms, Base 10 Logarithms with base 10 are called common logarithms.

**The subscript 10 is often dropped, so a log statement with no specified base is understood to be base 10.

EX #2: Evaluate the following logarithms and exponential expressions.

A . log 100

B . log √10

C . log1

1000

D .10log6 6

Ex #3: Evaluate these common logarithms with a calculator.

C.

Solving Simple Logarithmic Equations

• To solve an exponential equation, change it to a logarithmic equation.

• To solve a logarithmic equation, change it to an exponential equation.

Ex #4: Solve each equation by changing it to exponential form.

A . log 𝑥=3 B . log2𝑥 ¿5

Natural Logarithms, Base eNotation: The logarithmic function .

EX #5: Evaluate the following logarithmic and exponential expressions.

A.

B. C.

Evaluating Natural Logarithms with a Calculator

EX #6: Use a calculator to evaluate the logarithmic expressions.

A. = B.

C.

Independent Practice 3-3 : 2 to 36 even # - 20 minute

Key ConceptsLogarithmic function- the inverse of an exponential function.

By definition of logarithm, y = log4 x is the inverse of y = 4x.

Step 1: Graph y = 4x

Step 2: Draw y = x.

Step 3: Choose a few points on 4x. Reverse the coordinates and plot the points of y = log4 x.

EX #6: Graph y = log4 x

Ex# 7 :Graph y = log5 (x – 1) + 2.

Step 1: Graph y = log5 x

Step 2: Graph the function by shifting the points from the graph to the right 1 unit and up 2 units.

Basic function: or Graph: Analysis:

Graphing Logarithmic Functions

EX #8: Transforming Logarithmic Graphs

Describe how to transform the graph of = ln or = log into the graph 𝑦 𝑥 𝑦 𝑥of the given function.

D.

Independent Practice 3-3 : 38 to 56 even # - 30 minute

Home work 3-3 : 63- 68 allReview 3-3 notes do all problems again.