Name: _______________________
Period: ______________________
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:
2
3
4
5
6
7
8
Squares
4
9
16
25
36
49
64
Cubes
8
27
64
125
216
343
512
Fourth
16
81
256
625
1,296
2,401
4,096
Fifth
32
243
1,024
3,125
7,776
16,807
32,768
9
10
11
12
13
14
15
Squares
81
100
121
144
169
196
225
Cubes
729
1,000
1,331
1,728
2,197
2,744
3,375
Fourth
6,561
10,000
14,641
20,736
28,561
38,416
50,625
Fifth
59,049
100,000
161,051
248,832
371,293
537,824
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:
1
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
Example 7:
4x = 256
x = __________________
(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:
2
Example 11:
2(5x + 1) = 4(x + 7)
x = __________________
Example 12:
3(5x) = 9(x – 1)
x = __________________
Example 13:
5 x 
Example 14:
2(53 x ) 
1
25
1
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:
3
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 = _________________
Reflection:
Can there be another answer?
4
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
82 = 64
or
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:
9 3 
Example 13:
106 = 1,000,000
Reflection:
1
729
_____________________
_____________________
_____________________
5
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:
6
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:
7
Quotient Property of Logarithms
log b
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 b u x 
______________________
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:
8
Example 7:
log9 10 + log9 4 – log9 8
Example 9:
log6 (x + 7) – log6 (x + 5)
Example 8:
3 logb q – logb r
Example 11:
log3
Expand the expression.
Example 10:
log 4
2 11
5
Example 12:
log y
13 x
2
x4
y2
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:
Reflection:
ln e
Example 14:
ln e5
Example 15:
ln e2.1
9
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
log b u 
log10 u
log10 b
or
log u
log 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:
10
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 3:
log2 (4x – 3) = log2 (2x + 7)
Example 2:
log (2x – 12) = log (x + 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:
11
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:
12
Steps to Solve Exponential Equations
1.
2.
3.
4.
Take the log (or ln) of both sides.
Use the Power Property of logarithms to get the variable out of the exponent.
Divide both sides by the log on the LEFT (the one that is multiplying the x).
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:
13
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
x
-3
-2
-1
0
1
2
3
y
What is the horizontal asymptote? y = _____
( xh)
k
General form of an exponential function is f ( x)  b
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:
Reflection:
f(x) = 2x + 1
Example 3:
f(x) = 3x – 2
Example 4:
f(x) = 5(x + 3)
14
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:
Reflection:
f(x) = 5–x
What is the horizontal asymptote? y = _____
15
Do you remember e? e is approximately _________________________
Example 8:
f(x) = ex
Example 9:
What is the horizontal asymptote? y = _____
Example 10:
x
-3
-2
-1
0
1
2
3
Example 11:
Reflection:
1
f ( x)   
2
f(x) = e(x + 2) – 1
What is the horizontal asymptote? y = _____
x
What is the horizontal asymptote? y = _____
y
1
f ( x)   
3
( x  2)
x
Example 12:
1
f ( x)     1
7
16
The graph of an exponential function is given. Match the graph to one of the following functions.
Example 13:
A)
f ( x)  2 x 2  6
B) f ( x)  2( x 1)
C) f ( x)  3x 2  6
D) f ( x)  3  2( x 1)
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 14:
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 15:
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 16:
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.
Reflection:
17
Example 17:
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.
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 18: 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 19: 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:
18
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 f ( x)  k  logb ( x  h) or f ( x)  logb ( 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:
19
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:
Reflection:
ln (x – 3) + 5
Example 7:
log (x + 6) – 12
20