LOGARITHMIC FUNCTIONSSection 8.4

8.4 LOGARITHMIC FUNCTIONS

Objectives:

1. Write logarithmic functions in exponential form and back.

2. Evaluate logs with and without a calculator.

3. Evaluate logarithmic functions.

4. Understand logs and inverses.

5. Graph logarithmic functions.

Vocabulary:

logarithm, common logarithm, natural logarithm

In Section 8.3, we learned that if the interest of a bank account is 5% compounded, then the

total asset after t years is described by:

Yearly: At = P (1 + 0.05 )t

Monthly: At = P (1 + 0.05 / 12)12·t

Daily: At = P (1 + 0.05 / 365)365·t

Continuously: At = P e 0.05·t

In each case, as long as we know the time, t, we can calculate the final (total) asset:

Yearly: A5 = P (1 + 0.05 )5

Monthly: A10 = P (1 + 0.05 / 12)12·10

Daily: A2 = P (1 + 0.05 / 365)365·2

Continuously: A6 = P e 0.05·6

Now we would like to ask a reverse question:

How long does the initial deposit (investment) take to reach the target asset value?

Yearly: 2000 = 1200 (1 + 0.05 )t

LET’S THINK

LOGS TO THE RESCUE

O S W E G OINTRODUCING…

O S W E G Ohich

xponent

oes

n

EVALUATE THE EXPRESSIONS

Think: “Which exponent goes on 2 to give me 8?”3

2

3

0

Sorry, but “wego” does not really exist! In math, we use “logarithms.” The problems above would be written with the word “log” instead of “wego.”

EVALUATE THE EXPRESSIONS

4

2

-2

-3

6Which Exponent

Goes On…

SPECIAL LOGARITHM VALUES

Definition: Logarithm of y with base b

Let b and y be positive numbers, and b ≠ 1.

Then, logby = x if and only if y = bx.

Definition: Exponential Function

The function is of the form: f(x) = a · bx, where a ≠ 0, b > 0 and b ≠ 1.

REMEMBER THIS…?

REWRITING LOGARITHMIC EQUATIONS

   

Logarithmic Form Exponential Form

COMMON NOTATION

 

EVALUATING COMMON & NATURAL LOGS

 

 

Examples: Evaluate the common and natural logarithms.

a) log4

b) ln(1/5)

c) lne-3

d) log(1/1000)

0.602

-1.609

1

-3

Practice: Evaluate the common and natural logarithm.

a) ln0.25

b) log3.8

c) ln3

d) lne2007

0.845

-1.386

0.580

1

pg. 490 #16-19, 24-30, 36, 37

HOMEWORK

LOGARITHMIC FUNCTIONSSection 8.4 (Day 2)

RULE!

8.4 LOGARITHMIC FUNCTIONS

Objectives:

1. Write logarithmic functions in exponential form and back.

2. Evaluate logs with and without a calculator.

3. Evaluate logarithmic functions.

4. Understand logs and inverses.

5. Graph logarithmic functions.

Vocabulary:

logarithm, common logarithm, natural logarithm

From the definition of a logarithm, we noticed that the logarithmic function, g(x) = logbx, is the inverse of the exponential function f(x) = bx.

Recall:How do we verify if two functions are inverses?

WHAT DOES THIS MEAN?

This means that they offset each other, or they “undo” each other.

These two functions are inverses to each other.

USING INVERSES: SIMPLIFY THE EXPRESSION

x

x

x

x

USING INVERSES: SIMPLIFY THE EXPRESSION

x

2x

2x

3x

HOW DO WE FIND INVERSES?

1. Switch x and y.

2. Solve for y.

3. KAPOOYA! DONE!

4. Check using composition because we are diligent students.

In General…

LET’S LOOK AT THE SPECIFICS…

In General…  

1. Switch x and y.

2. Solve for y.

3. KAPOOYA! DONE!

4. Check using composition because we are diligent students.

LET’S LOOK AT THE SPECIFICS…

In General…  

1. Switch x and y.

2. Solve for y.

3. KAPOOYA! DONE!

4. Check using composition because we are diligent students.

Examples: Find the inverse of the function

a) y = log8x

b) y = ln(x – 3)

Answers:a) y = 8x

b) y = ex + 3

Practice: Find the inverse of

a) y = log2/5x

b) y = ln(2x – 10)

Answers:a) y = (2/5)x

b) y = (ex + 10)/2

Function FamilyThe graph of the function

y = f(x – h) k x – h = 0, x = h

is the graph of the functiony = f(x)

shift h unit to the right and k unit up/down.The graph of the function

y = f(x + h) k x + h = 0, x = –h

is the graph of the functiony = f(x)

shift h unit to the left and k unit up/down.

Logarithmic Function FamilyThe graph of the logarithmic function has the following characterisitcs:

y = logb(x − h) + k

1.) The line x = h is a vertical asymptote.

2.) The domain is x > h, and the range is all real numbers.

3.) If b > 1, the graph moves up to the right. If 0 < b < 1, the graph moves down to the right.

Example: Graph the function, state domain and range.

a) y = log1/2 (x + 4) + 2 b) y = log3(x – 2) – 1

1- 4

1 2

0

0

D: x > -4, R: all real numbers D: x > 2, R: all real numbers

NOTICE

Natural logs (ln) will be graphed in the same way. Just pick points from the table on your graphing

calculator.

Be careful! There is a difference between:

Vertical shift

Horizontal shift

pg. 491 #49-52, 58-63, 65-67

HOMEWORK