Warm – Up

•Practice worksheet 3.1•Practice identifying and using the correct formula which is necessary to solve a problem•Compound Interests and Annuities

C

LOGARITHMIC FUNCTIONS AND THEIR GRAPHS

Section 3.2

Objectives

• Students will be able to…• Decipher between and use each interest and exponential application formula• Determine the difference between a present value and future value annuity• Switch between exponential and logarithmic forms• (Evaluate logs and natural logs)

Definition of Logarithmic Function• For x > 0, a > 0 and a ≠ 1,

, if and only if

• The function given by

is called the logarithmic function with base a

• Logarithms are exponents ( is the exponent to which a must be raised to obtain x.

Switching Forms

• can be re – written as (3 to what power is 9?)

• can be re – written as

Practice

•“The Meaning of Logarithms” Worksheet•Problems 1 – 20

• _______ minutes

Closure – Exit Ticket

•On a note card(provided by Miss Young)•Write your name! (please)•Write the equation in exponential form: •Write the equation in logarithmic form:

•Hand your note – card to Miss Young on your way out the door•Have a great day!!

Warm – Up

•Write the equation in exponential form:

•Write the equation in logarithmic form:

Objectives

•Students will be able to…• Evaluate logs and natural logs•Use inverse symmetry to connect exponential and logarithmic graphs•Recognize and transform the parent graph of log functions

Evaluating Logarithms

1) , x = 32 2) , x = 1

3) , x = 2 4) , x =

Practice: Evaluate the Logarithm1. y = log216

2. y = log2( )

3. y = log416

4. y = log51

2

1

Common Logarithmic Function

• The logarithmic function with base 10

• Denoted: or just

• This is the log function on your calculator

• Example: Evaluate the function at each value of x

1) x = 10 2) x =

3) x = 2.5 4) x = -2

Properties of Logarithms1. loga 1 = 0 since a0 = 1.

2. loga a = 1 since a1 = a.

3. loga ax = x

4. If loga x = loga y, then x = y. one-to-one property

Examples:

1. Solve for x: log6 6 = x 2. Simplify: log3 35

Properties of Natural Logs• (Ln is the exact same idea as logs. Ln is in base e. Logs are (usually)

base 10)

1. ln 1 = 0 since e0 = 1.

2. ln e = 1 since e1 = e.

3. ln ex = x

4. If ln x = ln y, then x = y. one-to-one property

Examples: Simplify

2

1lne

eln3 1ln

Graphing Logarithmic Functions

• The graphs of inverse functions are reflections of each other in the line

• Sketch the graph of

• Sketch the graph of

Graphing Logarithmic Functions

• Plug in x – values that work nicely• (Can’t use calculator when not base 10)

• Example: Graph (state the changes, VA and x-intercept)

Graphing Logarithmic Functions

•Logarithmic Function:

• “a” – shrink or stretch graph (changes shape)• “h” – shifts the graph left or right (i.e. (x-2) shifts right 2)• “k” – shifts the graph up or down (i.e. x + 2 shifts up 2)

•Negative log flips graph over x–axis

Graphing Logarithmic Functions

• Example: Graph

Graphing Logarithmic Functions

• Example: Graph

Practice

•Work on “Graphing Logarithms” worksheet (finish for homework.

Closure

• Evaluate:

• Solve for x: log3 x = log3 (2x + 1)

• Graph: