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WARM UP
LOGARITHMIC FUNCTIONS
SWBAT identify key features and apply properties of logarithmic functions.
Given 2 MC and 2 CR problems, students will identify key features and apply properties of logarithmic functions with 80% accuracy.
Objective DOL
HOW ARE LOGARITHMS AND EXPONENTIALS RELATED?Essential Question
LOGARITHMIC TO EXPONENTIAL…
y = logbx
0 = log81
1 = 80
y = logbx
-4 = log2(1/16)
1/16 = 2-4
0 = log81 -4 = log2(1/16)
QUICK PRACTICE
QUICK PRACTICE
EXPONENTIAL TO LOGARITHMIC…
y = logbx
3 = log101000
y = logbx
1/2 = log93
103 = 1000 91/2 = 3
EVALUATING LOG AND LN ON THE CALCULATOR
Use ln (natural log (base e))
Use log button (common log (base 10))
There isn’t a base 5 button, so…
?
CHANGE OF BASE FORMULA
This will allow us to evaluate a logarithm with any base!
CHANGE OF BASE FORMULA
Practice
EVALUATE.
PROPERTIES OF LOGARITHMS
3log2log )32(log 101010
3log2log 3
2log 101010
3log 2 3log 102
10
Think-Pair-Share: Why do these look familiar? How can we remember them?
PROPERTIES OF LOGS/EXPONENTS
Think/Pair/Share – What do these properties have in common with Properties of Exponents? Explain your thinking.
.
PROPERTIES OF LOGS/EXPONENTS
Think/Pair/Share – What do these properties have in common with Properties of Exponents? Explain your thinking.
.
PROPERTIES OF LOGS/EXPONENTS
Think/Pair/Share – What do these properties have in common with Properties of Exponents? Explain your thinking.
CHANGE OF BASE/EXPAND/CONDENSE Practice rewriting several logarithmic
expressions using the properties (both expanding and collapsing):
WHICH PROPERTIES CAN YOU USE TO SIMPLIFY EACH?
REWRITE-EXPAND-CONDENSE PRACTICE
GIVEN LOG 3 =0.4771, LOG 4 = 0.6021, AND LOG 5 = 0.6990 Use the properties of logarithms to evaluate each
expression. Show your work for each step.
Example: log 12 log 12 = log 3(4) = log 3 + log 4
= 0.4772 + 0.6021= 1.0793
GIVEN LOG 3 =0.4771, LOG 4 = 0.6021, AND LOG 5 = 0.6990 Use the properties of logarithms to evaluate each
expression. Show your work for each step.
1. log 16 2. log 3/53. log 754. log 60
SOLVING EXPONENTIAL AND LOGARITHMIC EQUATIONS
Practice
APPLICATION
a) What property will be used to solve this equation? Will you expand or condense?
Power Property
APPLICATION
a) What property will be used to solve this equation? Will you expand or condense?
Power Property
APPLICATION
affectedpeople30N
25.62895N
2.255N 4
APPLICATION
a) What property will be used to solve this equation? Will you expand or condense?
Power Property
APPLICATION
a) What property will be used to solve this equation? Will you expand or condense?
Power Property
APPLICATION
Explain what happens in each step: Substitute in 300
Subtract 5 from both sides
Convert to log form
Change of base formula
Solution
APPLICATION
a) What property will be used to solve this equation? Will you expand or condense?
Power Property
WHAT IS A LOGARITHM? a number for a given base is the exponent to
which the base must be raised in order to produce the number
COMPLETE THE TABLE AND GRAPH THE EXPONENTIAL FUNCTION
WHAT ARE THE KEY FEATURES?
Domain:
Range:
Y-intercept:
X-intercept:
Asymptote:
End behavior:
All real numbers
All positive numbers; y > 0
(0, 1)
No x-intercept
y = 0
NOW GRAPH THE INVERSE
WHAT ARE THE KEY FEATURES?
Domain:
Range:
Y-intercept:
X-intercept:
Asymptote:
x > 0
All real number
No y-intercept
(1, 0)
x = 0
BACK TO THE INVERSE
HOW ARE LOGARITHMS AND EXPONENTIALS RELATED?Essential Question
DOL #1
DOL #2
DOL #3
Apply properties of logs to expand this logarithm and explain your reasoning.
DOL #4Maryville was founded in 1950. At that time, 500 people lived in the town. The population growth in Maryville follows the equation , where t is the number of years since 1950.
a)Determine when the population had doubled since the founding.
b) In what year was the population predicted to reach 25,000 people?
c) What social implications could the population growth in that number of years have on the town?
tP 5.1500
DOLMaryville was founded in 1950. At that time, 500 people lived in the town. The population growth in Maryville follows the equation , where t is the number of years since 1950.
a)Determine when the population had doubled since the founding. t = 15.327 years so 1965
b) In what year was the population predicted to reach 25,000 people? t = 24.926 so 1974.9Right before 1975c) What social implications could the population growth in that number of years have on the town?
tP 5.1500
Jobs, housing, schools, traffic, etc.