Download Logarithmic Functions

WARM UP
LOGARITHMIC FUNCTIONS
Objective

SWBAT identify key
features and apply
properties of
logarithmic functions.
DOL

Given 2 MC and 2 CR
problems, students
will identify key
features and apply
properties of
logarithmic functions
with 80% accuracy.
HOW ARE LOGARITHMS AND
EXPONENTIALS RELATED?
Essential Question
LOGARITHMIC TO EXPONENTIAL…
0 = log81
-4 = log2(1/16)

y = logbx

y = logbx

0 = log81

-4 = log2(1/16)

1 = 80

1/16 = 2-4
QUICK PRACTICE
QUICK PRACTICE
EXPONENTIAL TO LOGARITHMIC…
103 = 1000
91/2 = 3

y = logbx

y = logbx

3 = log101000

1/2 = log93
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
log10 (2  3)  log10 2  log10 3
2
log 10
 log 10 2  log 10 3
3
log 10 3
2
 2 log 10 3
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

1.
2.
3.
4.
Use the properties of logarithms to evaluate each expression.
Show your work for each step.
log 16
log 3/5
log 75
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
N  5  2.25
4
N  5  25.6289
N  30 people affected
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: All real numbers

Range: All positive numbers; y > 0

Y-intercept:

X-intercept:
(0, 1)
No x-intercept

Asymptote:
y=0

End behavior:
NOW GRAPH THE INVERSE
WHAT ARE THE KEY FEATURES?

Domain: x > 0

Range: All real number

Y-intercept:

X-intercept:
No y-intercept
(1, 0)

Asymptote:
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 #4
Maryville was founded in 1950. At that time, 500
people lived in the town. The population growth in
Maryville follows the equation P  500  1.5t , 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?
DOL
Maryville was founded in 1950. At that time,
500 people lived in the town. The population
growth in Maryville follows the equation
P  500  1.5 , where t is the number of years since
1950.
t
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 1974.9
Right before 1975
c) What social implications could the population
growth in that number of years have on the
town?
Jobs, housing, schools, traffic, etc.