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
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