Name: _______________________ Period: ______________________

Reflection: 1

Section 8.1 Solving Exponential Equations Objective(s): Solve exponential equations with the same base.

Essential Question: Explain why if two powers with the same base are equal, then their exponents are

equal.

Homework: Assignment 8.1 #1 – 18 in the homework packet.

Notes:

Squares Cubes Fourth Fifth

Squares Cubes Fourth Fifth

2 4 8 16 32

9 81 729 6,561 59,049

3 9 27 81 243

10 100 1,000 10,000 100,000

4 16 64 256 1,024

11 121 1,331 14,641 161,051

5 25 125 625 3,125

12 144 1,728 20,736 248,832

6 36 216 1,296 7,776

13 169 2,197 28,561 371,293

7 49 343 2,401 16,807

14 196 2,744 38,416 537,824

8 64 512 4,096 32,768

15 225 3,375 50,625 759,375

Review: What does x-1 mean?

If 54 = _______, then what does 5-4 equal? 5-4 = _________________

If 93 = _______, then what does 9-3 equal? 9-3 = _________________

If 152 = _______, then what does 15-2 equal? 15-2 = ________________

If two powers with the same base are equal, then their exponents are equal.

If bx = by, then x = y.

Solve the equation.

Example 1: 3x = 310 x = __________________

Example 2: 4(x + 3) = 42x x = __________________

Reflection: 2

Example 3: 3(7 – 2x) = 33 x = __________________

Example 4: 5(x + 10) = 5-2 x = __________________

Example 5: 25 + 3x = 2-4 x = __________________

What would happen if the base was not the same?

Example 6: 7x = 49 x = __________________

Example 7: 4x = 256 (see chart) x = __________________

You need to rewrite the base/answer so that the bases match. Then solve.

Example 8: 8(3x – 7) = 64 x = __________________

Example 9: 6(x - 8) = 364 x = __________________

Example 10: 3(2x + 3) = 9(2x – 1) x = __________________

Reflection: 3

Example 11: 2(5x + 1) = 4(x + 7) x = __________________

Example 12: 3(5x) = 9(x – 1) x = __________________

Example 13: 1

525

x x = __________________

Example 14: (5 3 ) 12

16

x x = __________________

What would happen if you can’t change one of the bases to match the other?

You need to rewrite BOTH sides using the same base number.

Example 15: 36x = 216 x = __________________

Example 16: 323x = 16(4x + 3) x = __________________

Example 17: 2(8 – 4x) = 1 x = __________________

(Hint: anything to the zero power is????)

Reflection: 4

Section 8.2 Logarithmic Functions Objective(s): Evaluate and simplify expressions using properties of logarithms.

Essential Question: Explain why you need to know the base to simplify a logarithm.

Homework: Assignment 8.2 #19 – 46 in the homework packet.

Notes:

5? = 125 Five raised to the ______________ power equals 125.

3? = 243 Three raised to the ______________ power equals 243.

Another way of saying the same thing is with logarithms (or log). Asking for the log of a number is asking

WHAT IS THE POWER?

Using the expression above, the log of 125 is ______________ and the log of 243 is ______________.

2? = 32 _________________ log 32 = _________________

4? = 1,024 _________________ log 1,024 = _________________

8? = 4,096 _________________ log 4,096= _________________

6? = 216 _________________ log 216 = _________________

log 6,561 = _________________

log 729 = _________________

log 16 = _________________ Can there be another answer?

Reflection: 5

So, to make it clear what power is correct, they write the BASE lower and smaller. So that there is only

ONE correct answer.

log4 16 = _________________ but log2 16 = _________________

What is the log 64? Can there be MANY different answers?

641 = 64 or 82 = 64 or 43 = 64 or 26 = 64

What makes the difference is the BASE.

log8 64 = _________________ The base is eight.

What is the base?

Example 1: log9 81 = 2 base = ___________________

Example 2: log8 516 = 3 base = ___________________

What is the power?

Example 3: log11 1,331 = 3 power = ___________________

Example 4: log4 1,024 = 5 power = ___________________

What is the ‘answer’?

Example 5: log7 49 = 2 answer = ___________________

Example 6: log3 27 = 3 answer = ___________________

Rewrite as an exponential equation.

Example 7: log5 625 = 4 _____________________

Example 8: log3 (1/9) = -2 _____________________

Example 9: log10 0.1 = -1 _____________________

Example 10: log1/4 64 = -3 _____________________

Rewrite as a logarithmic equation.

Example 11: 43 = 64 _____________________

Example 12: 3 1

9729

_____________________

Example 13: 106 = 1,000,000 _____________________

Reflection: 6

Find the value of the logarithmic expression. Evaluate.

Example 14: log4 64 = _____________________

Example 15: log3 81 = _____________________

Example 16: log7 16,807 = __________________

Example 17: log13 1/169 = __________________

Example 18: log10 1/1000 = _________________

Example 19: log1/2 16 = _____________________

Example 20: log1/4 256 = ____________________

Example 21: log14 14 = ______________________

Example 22: log3 3 = ________________________

Example 23: log5 1 = ________________________

Example 24: log15 1 = _______________________

Reflection: 7

Section 8.3 Properties of Logarithms Objective(s): Evaluate and simplify expressions using properties of logarithms.

Essential Question: Explain why the logarithm of a negative number is undefined.

Homework: Assignment 8.3 #47 – 66 in the homework packet.

Notes:

log2 4 + log2 8 = ______ + ______ = ______ log2 32 = _________________________

So, log2 4 + log2 8 = log2 32

How does log2 4 + log2 8 make log2 32 ???? ________________________________________

log4 4 + log4 16 = ______ + ______ = ______ log4 64 = _________________________

So, log4 4 + log4 16 = log4 64

How does log4 4 + log4 16 make log4 64 ???? _______________________________________

Using the same idea…

log4 7 + log4 9 = _______________________ log5 6 + log5 2 + log5 3 = _______________________

Product Property of Logarithms

logb u v ______________________

log2 8 – log2 4 = ______ – ______ = ______ log2 2 = _________________________

So, log2 8 – log2 4 = log2 2

How does log2 8 – log2 4 make log2 2???? ________________________________________

log7 22 – log7 2 = _______________________ log6 4 + log6 5 – log6 2 = _____________________

Reflection: 8

Quotient Property of Logarithms

logb

u

v ______________________

log2 2 + log2 2 + log2 2 = log2 ____

There are THREE log2 2’s

So, 3log2 2 = log2 8. How can you make the answer, 8, on the left? _________________________

The 3 goes where? ____________________________________

log2 4 + log2 4 + log2 4 + log2 4 + log2 4 = log2 ____

There are FIVE log2 4’s

So, 5log2 4 = log2 1024. How can you make the answer, 1024, on the left? _________________________

The 5 goes where? ____________________________________

2log3 5 = log3 ____ 4log5 2 = log5 ____ -2log3 7 = log3 ____

Power Property of Logarithms

log x

b u ______________________

Condense the expression.

Example 1: log2 9 + log2 6 Example 2: log7 12 + log7 x

Example 3: log9 15 – log9 8 Example 4: log5 x – log5 y

Example 5: log2 7-4 Example 6: logw pr

Reflection: 9

Example 7: log9 10 + log9 4 – log9 8 Example 8: 3 logb q – logb r

Example 9: log6 (x + 7) – log6 (x + 5)

Expand the expression.

Example 10: 4

2 11log

5 Example 12:

13log

2y

x

The number e is sometimes called Euler's number after the Swiss mathematician Leonhard Euler

(pronounced OILER). It is an irrational number, like , and is approximately 2.718.

loge x is more commonly written as ln x. ln x is called the natural logarithm (logarithmus naturalis)

loge x is a "natural" log because it appears so often in mathematics.

Find the exact value.

Example 13: ln e Example 14: ln e5 Example 15: ln e2.1

Reflection: 10

What does log 7 mean? What is the base if they don’t write one in?

The base is ALWAYS 10!

The common logarithm is the logarithm with base 10.

log10 x is more commonly written as log x.

Special Rules

log 10 = 1 log 1 = 0 ln e = 1 ln 1 = 0

Occasionally, we need to know the approximate value of logs that can’t be found on the chart. For

example, log2 7 is what? Seven is NOT a power of two. Your next thought might be to use a calculator.

No calculators in the past (and few now) can calculate log2 7. For this reason, there is a formula called

the change of base formula. It allows you to change the log into something that can be entered into a

calculator. (The log button on your calculator is log10)

Change of Base Formula

10

10

log loglog or

log logb

u uu

b b

Use the change of base formula to rewrite the expression.

Example 16: log9 2

Example 17: log1/3 12

Example 18: log5 1/18

Reflection: 11

Section 8.4 Solving Exponential and Logarithmic Equations Objective(s): Solve exponential and logarithmic equations.

Essential Question: Explain the purpose of taking the log of both sides of an exponential equation.

Homework: Assignment 8.4 #67 – 79 in the homework packet.

Notes:

If two logarithms with the same base are equal, then their ‘answers’ are equal.

If logb x = logb y, then x = y.

Solve the equation.

Example 1: log6 x = log6 21

Example 2: log (2x – 12) = log (x + 7)

Example 3: log2 (4x – 3) = log2 (2x + 7)

Example 4: log3 (x + 2) = 2log3 4

Sometimes you have to solve equations with TWO logs on one side. In this case, you must use a

Logarithmic Property to condense the two logs into one log.

Example 5: log 3x = log 5 + log (x – 2) x = _____________________

Example 6: log3 (x – 2) = log3 25 + log3 (x – 4) x = _____________________

Reflection: 12

Some logarithmic equations have a log on one side and a NUMBER on the other. These equations are

solved by rewriting them in exponential form.

Example 7: log2 x = 5 x = _____________________

Example 8: log3 (x + 1) = 2 x = _____________________

Some logarithmic equations have TWO logs on one side and a NUMBER on the other. In this case, you

must use a Logarithmic Property to condense the two logs into one log and then rewrite the equation in

exponential form

Example 9: log9 5 + log9 x = 1 x = _____________________

Example 10: ln 8 + ln x = 0 x = _____________________

In the first section of this packet, we solved exponential equations that had the same base or could be

written with the same base.

7(x + 2) = 343 x = _____________________

What if you had an exponential equation that could NOT be written with the same base?

7x = 5

Reflection: 13

Steps to Solve Exponential Equations

1. Take the log (or ln) of both sides.

2. Use the Power Property of logarithms to get the variable out of the exponent.

3. Divide both sides by the log on the LEFT (the one that is multiplying the x).

4. Get x by itself.

Solve the equation. Give an exact answer.

Example 11: 7x = 5 x = _____________________

Example 12: 53x = 4.9 x = _____________________

Example 13: e4x = 2 x = _____________________

Example 14: 5(x + 8) = 7 x = _____________________

Example 15: e(x – 1) = 7 x = _____________________

Reflection: 14

Section 8.5 Exponential Growth and Decay Objective(s): Graph exponential functions. Develop mathematical models using exponential equations.

Essential Question: Explain how the irrational number e can be used in the ‘real world’.

Homework: Assignment 8.5 #80 – 96 in the homework packet.

Notes:

Graph the exponential function.

Example 1: f(x) = 2x What is the horizontal asymptote? y = _____

x y

-3

-2

-1

0

1

2

3

General form of an exponential function is ( )( ) x hf x b k

Where the k shifts the graph ______________ and ______________,

and h shifts the graph ______________ and ______________,

NOTICE: The graph does not go through the origin (0, 0). Instead, it goes through the point (0, 1), and

the horizontal asymptote is 1 unit below that point.

Example 2: f(x) = 2x + 1 Example 3: f(x) = 3x – 2 Example 4: f(x) = 5(x + 3)

Reflection: 15

Example 5: f(x) = 4(x - 2) + 3 What is the horizontal asymptote? y = _____

What happens when you have a negative ___________________________ ?

Example 6: f(x) = – (2x) What is the horizontal asymptote? y = _____

How about if the negative is on the variable _________________________ ?

Example 7: f(x) = 5–x What is the horizontal asymptote? y = _____

Reflection: 16

Do you remember e? e is approximately _________________________

Example 8: f(x) = ex

Example 9: f(x) = e(x + 2) – 1

What is the horizontal asymptote? y = _____ What is the horizontal asymptote? y = _____

Example 10: 1

( )2

x

f x What is the horizontal asymptote? y = _____

x y

-3

-2

-1

0

1

2

3

Example 11:

( 2)1

( )3

x

f x Example 12: 1

( ) 17

x

f x

Reflection: 17

Exponential Growth Models

When a real-life quantity increases by a fixed percent each year, the amount can be modeled by the

following equation:

y = a(1 + r)t where a is the initial amount, r is the percent increase, and t is the time in years.

When a real-life quantity decreases by a fixed percent each year, the amount can be modeled by the

following equation:

y = a(1 – r)t where a is the initial amount, r is the percent decrease, and t is the time in years.

Example 13: In 1990, the cost of tuition at a state university was $4300. During the next eight years,

the tuition rose 4% each year. Write a model giving the cost of tuition.

Example 14: You buy a new car for $24,000. Each year, the value of the coin decreases by 16%. Write

a model giving the value of the car.

Example 15: In 1980, about 2 million US workers worked at home. During the next ten years, the

number of workers working at home increased by 5% each year. Write a model giving the number of

workers (in millions) working at home.

Example 16: You drink a beverage with 120 milligrams of caffeine. Each hour, the amount of caffeine

in your system decreases by 12%. Write a model giving the amount of caffeine in your system.

Reflection: 18

When a real-life quantity doubles in a fixed time length, the amount can be modeled by the following

equation:

y = a(2)t/k where a is the initial amount, t is the time, and k is the doubling period.

When a real-life quantity is cut in half (half-life) in a fixed time length, the amount can be modeled by

the following equation:

y = a(1/2)t/k where a is the initial amount, t is the time, and k is the half-life period.

What would be the equation for when a quantity triples? ________________________________

Example 17: A population doubles every 9 years. If there are 300 deer to begin with, write a growth

model to show the number of deer in t years.

Example 18: The half-life of element X is 12.2 years. If there are 200 grams of the element, write a

growth model to show the amount of the element in t years.

Reflection: 19

Section 8.6 Graphing Logarithmic Functions Objective(s): Graph logarithmic functions.

Essential Question: How are the graphs of exponential functions related to graphs of logarithmic

functions?

Homework: Assignment 8.6 #97 – 114 in the homework packet.

Notes:

Graph the logarithmic function.

Example 1: f(x) = log2 x What is the vertical asymptote? x = _____

x y

1/16

1/8

1/4

1/2

1

2

4

8

General form of a logarithmic function is ( ) log ( ) or ( ) log ( )b bf x k x h f x x h k

Where the k shifts the graph ______________ and ______________,

and h shifts the graph ______________ and ______________,

The basic graph goes through the point (1, 0) with a vertical asymptote 1 unit to the left of the point.

Example 2: f(x) = log3 (x – 1) Example 3: f(x) = 2 + log4 x

What is the vertical asymptote? x = _____ What is the vertical asymptote? x = _____

Reflection: 20

Example 4: f(x) = -1 + log9 (x + 2)

Example 5: f(x) = ln x

What is the vertical asymptote? x = _____ What is the vertical asymptote? x = _____

Describe how to transform the graph.

Example 6: ln (x – 3) + 5 Example 7: log (x + 6) – 12