Warm – Up Practice worksheet 3.1 Practice identifying and using the correct formula which is...
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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: