Algebra II
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4-3 Study Guide
Logarithmic Functions
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Attendance Problems. Use mental math (no calculator, do all of your work
mentally) to evaluate the following.
1. ! 4
−3
2. ! 16
1
4
3. ! 10
−5
⎛ 2⎞
4. ! ⎜ ⎟
⎝ 3⎠
−3
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2. A power has a base of –2 and exponent of 4. Write and evaluate the power.
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• I can write equivalent forms for exponential and logarithmic functions.
• I can write, evaluate, and graph logarithmic functions.
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Vocabulary
logarithm
common logarithm logarithmic function
Common Core
• CCSS.MATH.CONTENT.HSA.CED.A.2 Create equations in two or more variables to
represent relationships between quantities; graph equations on coordinate axes with labels and
scales.
• CCSS.MATH.CONTENT.HSA.CED.A.3 Represent constraints by equations or inequalities,
and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable
options in a modeling context. For example, represent inequalities describing nutritional and
cost constraints on combinations of different foods.
• CCSS.MATH.CONTENT.HSF.IF.C.7 Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and using technology for more complicated
cases.*
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Algebra II
4-3 Study Guide
Page 2! of 11
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How many times would you have to double $1 before you had $8? You could use
an exponential equation to model this situation. 1(2x) = 8. You may be able to solve
this equation by using mental math if you know 23 = 8. So you would have to
double the dollar 3 times to have $8.
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How many times would you have to double $1 before you had $512? You could
solve this problem if you could solve 2x = 8 by using an inverse operation that
undoes raising a base to an exponent equation to model this situation. This
operation is called finding the logarithm. A logarithm is the exponent to which a
specified base is raised to obtain a given value.
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You can write an exponential equation as a logarithmic equation and vice versa.
Video Example 1. Write each exponential equation in logarithmic form.
A. ! 5 3 = 125
!
!
D. ! 10
!
!
−2
= 0.01
B. ! 61 = 6
E. ! 4 x = 16
C. ! 9 0 = 1
E
b4-3
> 0,
b ≠ 1Guide
Study
Algebra II
1
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Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
Exponential
Equation
Logarithmic
Form
a.
2 6 = 64
log 2 64 = 6
b.
41 = 4
log 4 4 = 1
The exponent is the logarithm.
c.
50 = 1
log 5 1 = 0
Any nonzero base to the 0 power is 1.
d.
5 -2 = 0.04
log 5 0.04 = -2
e.
3 x = 81
log 3 81 = x
The base of the exponent becomes the
base of the logarithm.
An exponent (or log) can be negative.
The log (and the exponent) can be a
variable.
Write each exponential equation in logarithmic form.
Example 1. Write
in logarithmic0 form.
2 each exponential equation
3
!
!
A.! 35 = 2431a.
D. ! 6 −1 =
!
!
1
6
9 = 81 B. ! 10 0 =1b.
3 = 27
1
E. ! a b = c
x =
4
C. ! 101c.
= 10,000
1(x ≠ 0)
4-3 Logarithmic Functions
249
Guided Practice. Write each exponential equation in logarithmic form.
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3. ! 9 2 = 81
4. ! 33 = 27
5. ! x 0 = 1 ( x ≠ 0 )
Video Example 2. Write each logarithmic form in exponential equation.
• ! log 5 25 = 2
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!
D. ! log 3 = 1
!
!
3
B. ! log 2 8 = 3
E. ! log 8 1 = 0
C. ! log 5 0.2 = −1
5/4/11 3:18:1
LE
Algebra II
2
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4-3 Study Guide
Converting from Logarithmic to Exponential Form
Write each logarithmic equation in exponential form.
Logarithmic
Equation
a.
log 10 100 = 2
b.
log 7 49 = 2
c.
Exponential
Form
The base of the logarithm becomes
the base of the power.
10 2 = 100
7 2 = 49
log 8 0.125 = -1
The logarithm is the exponent.
A logarithm can be a negative
number.
8 -1 = 0.125
d.
log 5 5 = 1
51 = 5
e.
log 12 1 = 0
12 0 = 1
Example 2. Write
logarithmic
formequation
in exponential
equation. form.
Writeeach
each
logarithmic
in exponential
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A. ! log 9 9 = 1
D. ! log 4
!
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B. ! log 2 512 = 9
2a. log 10 10 = 1
-3
1
= −2
16
C. ! log 8 64 = 2
2b. log 12 144 = 2 2c. log _1 8 =
2
E. ! log b 1 = 0
A logarithm is an exponent, so the rules for exponents also apply to logarithms.
You may have noticed the following properties in the last example.
Guided Practice. Write each logarithmic form in exponential equation.
6. ! log10 10 = 1
7. ! log12 144 = 2
Special Properties of Logarithms
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8. ! log 1 8 = −3
2
For any base b such that b > 0 and b ≠ 1,
A logarithm is an exponent, so the rules for exponents also apply to logarithms.
LOGARITHMIC
EXPONENTIAL
FORM
EXAMPLE
You
may have noticedFORM
the following
properties in the last
example.
Logarithm of Base b
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!
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log bb = 1
b1 = b
log 1010 = 1
10 1 = 10
b0 = 1
log 101 = 0
10 0 = 1
Logarithm of 1
log b1 = 0
Special
ofStudy
Logarithms
AlgebraProperties
II
4-3
Guide
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For any base b such that b > 0 and b ≠ 1,
LOGARITHMIC FORM
EXPONENTIAL FORM
E
b1 = b
lo
1
b0 = 1
lo
1
Logarithm of Base b
log bb = 1
Logarithm of 1
log b1 = 0
A logarithm with base 10 is called a common logarithm. If no base is written for a
logarithm, the base is assumed to be 10. For example, log 5 = log105.
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A logarithm
withmath
base
10 is some
called
a common logarithm . If no b
You can use mental
to evaluate
logarithms.
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a logarithm,
the base is assumed to be 10. For example, log 5 = l
Video Example 3. Evaluate by using mental math.
A. log 100
B. ! log 3
1
9
You can use mental math to evaluate some logarithms.
3
Evaluating Logarithms by Using Mental Math
Evaluate by using mental math.
A log 1000
1
B log 4 _
10 ? = 1000
The log is the exponent.
10 3 = 1000
Think: What power of
the base is the value?
log 1000 = 3
4
1
4? = _
4
1
4 -1 = _
4
1 = -1
log 4 _
4
Algebra II
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4-3 Study Guide
Example 3. Evaluate by using mental math.
log 0.01
B. ! log 5 125
C. ! log 5
!
!
1
5
Guided Practice. Evaluate by using mental math.
9. log 0.00001
10. ! log 25 0.04
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Because logarithms are the inverses of exponents, the
inverse of an exponential function, such as y = 2x, is a
logarithmic function, such as y = log2x.
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You may notice that the domain and range of each function
are switched.
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The domain of y = 2x is all real numbers (R), and the range
is {y|y > 0}. The domain of y = log2x is {x|x > 0}, and the
range is all real numbers (R).
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Video Example 4. Use x = -2, -1, 0, 1, and 2 to
graph the function ! f ( x ) = 4 x. Then graph its inverse.
Describe the domain and the range of the inverse
function.
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!
E
The domain of y = 2 is all real numbers (!), and
⎧
⎫
the range is ⎨ y | y > 0 ⎬. The domain of y = log 2 x is
⎧
⎫ ⎩
⎭
> 0 ⎬, andIIthe range is all real
(!).
⎨ x | x Algebra
4-3numbers
Study Guide
⎩
⎭
4
-4
4
8
12
-4
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Graphing Logarithmic Functions
Use the given x-values to graph each function. Then graph its inverse.
Describe the domain and range of the inverse function.
A f (x) = 3 x; x = -2, -1, 0, 1, and 2
Graph f (x) = 3 x by using a table of values.
x
-2 -1
1
_
9
f(x) = 3 x
1
_
3
0
1
2
1
3
9
y
8
1
_
9
f -1(x) = log 3 x
f(x) = 3x
6
To graph the inverse, f -1(x) = log 3 x,
reverse each ordered pair.
x
(2, 9)
4
f -1(x) = log3 x
2
1
_
3
1
3
9
-2 -1
0
1
2
(9, 2)
-2
2
4
6
x
8
-2
⎧
⎫
The domain of f -1(x) is ⎨ x | x > 0 ⎬, and the range is !.
⎩
⎭
B f (x) = 0.8 x; x = -3, 0, 1, 4, and 7
Graph f (x) = 0.8 x by using a table of values.
Round the output values to the nearest
tenth, if necessary.
x
-3
0
f(x) = 0.8 x
2
1
1
4
8
7
y
-1
f (x) = log 0.8 x
6
0.8 0.4 0.2
4
-1
To graph f (x) = log 0.8 x, reverse each
ordered pair.
x
2
f -1(x) = log 0.8 x -3
1
0
0.8 0.4 0.2
1
4
7
2
f(x) = 0.8 x
-2
2
4
-2
⎧
⎫
The domain of f -1(x) is ⎨ x | x > 0 ⎬, and the range is !.
⎩
⎭
x
()
4. Use x = -2, -1, 1, 2, and 3 to graph f (x) = __34 . Then
graph its inverse. Describe the domain and range of the
6
x
8
Algebra II
4-3 Study Guide
Example 2. Use the x-values {–2, –1, 0,
1, 2}. Graph the function and its
inverse. Describe the domain and
range of the inverse function.
A. ! f ( x ) = 1.25 x
!
!
!
!
!
!
!
!
1
B. ! f ( x ) = ⎛⎜ ⎞⎟
⎝ 2⎠
!
x
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Algebra II
4-3 Study Guide
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11. Guided Practice. Use x = –2, –1, 1, 2, and 3 to
x
3
graph ! f ( x ) = ⎛⎜ ⎞⎟ . Then graph its inverse. Describe
⎝ 4⎠
the domain and range of the inverse function.
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Helpful Hint
The
key is used to evaluate logarithms in
base 10.
inverse of log.
!
is used to find 10x, the
Algebra II
PLE
se
s
the
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4-3 Study Guide
Environmental Application
Chemists regularly
test rain samples to
determine the rain’s
acidity, or concentration
of hydrogen ions (H +).
Acidity is measured
in pH, as given by
the function pH = log ⎡⎣ H + ⎤⎦, where ⎣⎡ H + ⎤⎦
represents the hydrogen
ion concentration in
moles per liter.
Hydrogen Ion Concentration
of Rainwater
Central North Dakota
0.0000009 mol/L
Eastern Ohio
0.0000629 mol/L
Central California
0.0000032 mol/L
Find the pH of rainwater
from each location.
Central New Jersey
0.0000316 mol/L
Eastern Texas
0.0000192 mol/L
A Central New Jersey
The hydrogen ion concentration is 0.0000316 moles per liter.
pH = - log ⎡⎣H +⎤⎦
pH = - log (0.0000316) Substitute the known values in the function.
Use a calculator to find the value of the
key.
logarithm in base 10. Press the
The rainwater has a pH of about 4.5.
B Central North Dakota
The hydrogen ion concentration is 0.0000009 moles per liter.
pH = - log ⎡⎣H +⎤⎦
pH = - log (0.0000009) Substitute the known values in the function.
Use a calculator to find the value of the
key.
logarithm in base 10. Press the
The rainwater has a pH of about 6.0.
5. What is the pH of iced tea with a hydrogen ion concentration
of 0.000158 moles per liter?
Algebra II
4-3 Study Guide
Example 5. The table lists the hydrogen ion
concentrations for a number of food items.
Find the pH of each.
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Substance
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H+ conc. (mol/L)
Milk
0.00000025
Tomatoes
0.0000316
Lemon juice
0.0063
12. Guided Practice. What is the pH of iced tea with a hydrogen ion concentration
of 0.000158 moles per liter?
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4-3 Assignment (p 253) 17-28; 39-42.