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.