Tuesday Bellwork

•Pair-Share your homework from last night •We will review this briefly

•Remember, benchmark tomorrow

This week:• Monday: Logarithmic Functions and Their Graphs• Tuesday: “ “ cont. • Wednesday: Benchmark• Thursday: Properties of Logarithms• Friday: Review/Quiz OR Solving Exponential with Logarithms

The standard• F.BF.5. Understand the inverse relationship between

exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Logarithmic Functions and Their Graphs

Properties of logarithmsSection 3.3

Consider

This is a one-to-one function, therefore it has an inverse.

The inverse is called a logarithm function.

Example: Two raised to what power is 16?

The most commonly used bases for logs are 10:

and e:

is called the natural log function.

is called the common log function.

Definition of Logarithmic Function

b > 0; b 1

Logarithmic Form Exponential Form

y = logb x x = by

The log to the base “b” of “x” is the exponent to which “b” must

be raised to obtain “x”

y = log10 x

y = log e x

x = 10 y

x = e y

Section I on HW

Change from Logarithmic To Exponential Form

Log 2 8 = 3 8 = 23

5 = 25 ½Log 25 5 = ½

Change from Exponential To Form Logarithmic

49 = 7 2 log 7 49 = 2

1/5 = 5 –1 log 5 (1/5) = -1

Section II on HW

Using the Definition of log!

1. log3 x = 4

2. log100.01 = x

3. logx49 = 2

x = 34 = 81

0.01 = 10x

10-2 = 10x x = -2

49 = x2 x=7

Section III on HW

YOU use the definition of log:

Write each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

Solution With the fact that y = logb x means by = x,

c. log3 7 = y or y = log3 7 means 3y = 7.

a. 2 = log5 x means 52 = x.Logarithms are exponents.Logarithms are exponents.

b. 3 = logb 64 means b3 = 64.Logarithms are exponents.Logarithms are exponents.

Section III on HW

How to evaluate expressions: Pre-Calc Cookbook:

1.Set expression equal to y.

2.Identify the ‘b’ & ‘x’

3.Use the formula to convert to exponential form.

4.Make common bases (if not already)

5.Solve for y.

Similar to HW section IV

Example: Pre-Calc Cookbook:

1.Set expression equal to y.

2.Identify the ‘b’ & ‘x’

3.Use the formula to convert to exponential form.

4.Make common bases (if not already)

5.Solve for y.

a. log2 16 b. log3 9 c. log25 5

Solution

log25 5 = 1/2 because 251/2 = 525 to what power is 5?c. log25 5

log3 9 = 2 because 32 = 93 to what power is 9?b. log3 9

log2 16 = 4 because 24 = 162 to what power is 16?a. log2 16

Logarithmic Expression Evaluated

Question Needed for Evaluation

Logarithmic Expression

YOU Evaluate the expressions:

Properties of Logarithmic FunctionsIf b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Log15 1 = 0

Log10 10 = 1

Log5 5x = x

3log x = x 3

150 = 1

101 = 10

5x = 5x

Properties of Common Logarithms

General Properties Common Logarithms

1. logb 1 = 0 1. log 1 = 0

2. logb b = 1 2. log 10 = 1

3. logb bx = 0 3. log 10x = x4. b logb x = x 4. 10 log x = x

Examples of Logarithmic Properties

log 4 4 = 1

log 8 1 = 0

3 log 3 6 = 6

log 5 5 3 = 3

2 log 2 7 = 7

Properties of Natural Logarithms

General Properties Natural Logarithms

1. logb 1 = 0 1. ln 1 = 0

2. logb b = 1 2. ln e = 1

3. logb bx = 0 3. ln ex = x4. b logb x = x 4. e ln x = x

Examples of Natural Logarithmic Properties

e log e 6 = e ln 6 = 6

log e e 3 = 3

Standard Based Questions:

Use the inverse properties to simplify:

27 ln 4

ln

7.1 log

1. ln 2.

3. 4. log1000

5. log10 6. 10

x x

x

e

e e

e

Section V on HW

Tuesday Independent Practice

Logarithm Functions HW3: •Complete Sections I, II, III, IV, & V.

WEDNESDAYBENCHMARK!!!

ThursdayThursday

Thursday Bellwork Answer the following questions on a separate

piece of paper that you can turn in?

What did you think of the benchmark?

What do we need more practice on? What do we have mastery of?

Characteristics of the Graphs of Logarithmic Functions of the Form f(x)

= logbx The x-intercept is 1. There is no y-intercept.

The y-axis is a vertical asymptote. (x = 0)

If 0 < b < 1, the function is decreasing. If b > 1, the function is increasing.

The graph is smooth and continuous. It has no sharp corners or edges.

-2 -1

6

2 3 4 5

5

4

3

2

-1

-2

6

f (x) = logb xb>1

-2 -1

6

2 3 4 5

5

4

3

2

-1

-2

6

f (x) = logb x0<b<1

Since logs and exponentials are inverses the domain and range switch!…the x values and y values are exchanged…

Graph and find the domain of the following functions.

y = ln x

x y

-2-101234

.5

cannot takethe ln of a (-) number or 0

0ln 2 = .693ln 3 = 1.098ln 4 = 1.386

ln .5 = -.693

D: x > 0

f

x y = 2 x

–31

8

–21

4

–11

2

0 1

1 2

2 4

3 8

f –1

x = 2 y

1

8 –3 1

4 –2 1

2 –1

1 0

2 1

4 2

8 3

Ordered pairs reversed

y

x

y

5 10 –5

5

10

–5

f -1

x = 2y

or y = log2x

f y = 2x

y = x

DOMAIN of = (– , ) = RANGE of

RANGE of f = (0, ) = DOMAIN of

Logarithmic Function with Base 2

f

f -1

f -1

Using Calculator to Evaluate: ln(10)

> Calculate

> ‘ctrl’ then ‘ex’ => ln( )

> ’10’ => ln(10)

> ‘enter’ => ln(10)

> ‘menu’

> ’2: Number’

> ‘1: Convert to Decimal’ => Ans>Decimal

> ‘enter’ => 2.30259

We Evaluate: ln(12.4)

> Calculate

> ‘ctrl’ then ‘ex’ => ln( )

> ’10’ => ln(12.4)

> ‘enter’ => ln(12.4)

> ‘menu’

> ’2: Number’

> ‘1: Convert to Decimal’ => Ans>Decimal

> ‘enter’ => 2.5177

YOU Calculator to Evaluate:

1. ln(45) =

2. ln(0.234) =

3. ln(-3.45) =

1. = 3.80666

1. = -1.45243

2. = non-real number

Similar to section VI in HW

Homework: Complete ALL Logarithm Functions HW 3

FRIDAY BELLWORK

Copyright © Cengage Learning. All rights reserved.

3.3 Properties of Logarithms

What You Should Learn

• Rewrite logarithms with different bases.

• Use properties of logarithms to evaluate or rewrite logarithmic expressions.

• Use properties of logarithms to expand or condense logarithmic expressions.

• Use logarithmic functions to model and solve real-life problems.

Properties of Logarithms

Properties of Logarithms

Example 1 – Using Properties of Logarithms

Write each logarithm in terms of ln 2 and ln 3.

a. ln 6 b. ln

Solution:

a. ln 6 = ln(2 3)

= ln 2 + ln 3

b. ln = ln 2 – ln 27

= ln 2 – ln 33

= ln 2 – 3 ln 3

Rewrite 6 as 2 3.

Product Property

Quotient Property

Rewrite 27 as 33

Power Property

Rewriting Logarithmic Expressions

Rewriting Logarithmic Expressions

The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is true because they convert complicated products, quotients, and exponential forms into simpler sums, differences, and products, respectively.

Example 2 – Expanding Logarithmic Expressions

Use the properties of logarithms to expand each expression.

a. log45x3y

b. ln

Solution:

a. log45x3y = log45 + log4x3 + log4 y

= log45 + 3 log4x + log4y

Product Property

Power Property

Example 2 – Solution

Rewrite radical usingrational exponent.

Power Property

Quotient Property

cont’d

Rewriting Logarithmic Expressions

In Example 5, the properties of logarithms were used to expand logarithmic expressions.

In Example 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic expressions.

Example 3 – Condensing Logarithmic Expressions

Use the properties of logarithms to condense each expression.

a. log10x + 3 log10(x + 1)

b. 2ln(x + 2) – lnx

c. [log2x + log2(x – 4)]

Example 3 – Solutiona. log10x + 3 log10(x + 1) = log10x1/2 + log10(x + 1)3

b. 2 ln(x + 2) – ln x = ln(x + 2)2 – ln x

Power Property

Product Property

Quotient Property

Power Property

Example 3 – Solution

c. [log2x + log2(x – 4)] = {log2[x(x – 4)]}

= log2[x(x – 4)]1/3

cont’d

Power Property

Product Property

Rewrite with a radical.

Homework: Properties of Logs HW4

Monday, March 23, 2015

• F.BF.5. Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

3.4 Exponential and Logarithmic Equations

One-to-One Properties

Inverse Property

One-to-One Properties

If x 6 = x y , then 6 = y

If ln a = ln b, then a = b

Inverse Property

Given e x = 8; solve for xTake the natural log of each side.

ln e x = ln 8Pull the exponent in front

x ( ln e) = ln 8(since ln e = 1)

x = ln 8

Solve for x

3 x = 64

take the natural log of both sides

ln 3 x = ln 64

x( ln 3) = ln 64

x = ln 64 = 3.7855..

ln 3

Solve for x

e x – 8 = 70

Solve for x

e x – 8 = 70

e x = 78

ln e x = ln 78

x = ln 78

x = 4.3567..

Solve for a

( ¼ ) a = 64

Solve for K

Log 5 K = - 3

Solve for x

2 x – 3 = 32

Solve for x

3501

400

xe

Solve for x

e 2.724x = 29

Solve for a

ln a + ln ( a + 3) = 1

Will need the quadratic formula

Solve for x one more time

e 2x – e x – 12 = 0

factor

Solve for x one more time

e 2x – e x – 12 = 0

factor

(e x – 4)(e x + 3 ) = 0

So e x – 4 = 0 or e x + 3 = 0

e x = 4 e x = - 3

x = ln 4 x = ln -3

Homework:Solving Exponential Equations with Logarithms HW5