lecture 11 sections 4.3-4.4 logarithmic functions
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MATH 108
Section 4.3
Logarithmic Functions
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EXAMPLE 1 Converting from Exponential to Logarithmic Form
Write each exponential equation in logarithmic form.
Solution
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EXAMPLE 2 Converting from Logarithmic Form to Exponential Form
Write each logarithmic equation in exponential form.
Solution
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3 2
1a log 81 b log
8
(a) 3 raised to what power yields 81?
(b) 2 raised to what power yields ?1
8
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EXAMPLE 3 Evaluating Logarithms
Find the value of each of the following logarithms.
Solution
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COMMON LOGARITHMS
1. log 10 = 1
2. log 1 = 0
3. log 10x = x
The logarithm with base 10 is called the common logarithm and is denoted
by omitting the base: log x = log10
x. Thus,
y = log x if and only if x = 10 y.
Applying the basic properties of logarithms
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NATURAL LOGARITHMS
1. ln e = 1
2. ln 1 = 0
3. log ex = x
The logarithm with base e is called the natural logarithm and is denoted by
ln x. That is, ln x = loge
x. Thus,
y = ln x if and only if x = e y.
Applying the basic properties of logarithms
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Natural Logarithm Function
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EXAMPLE 7 Using Transformations
Start with the graph of f (x) = log
3
x and use transformations to sketch the
graph of each function.
State the domain and range and the vertical asymptote for the graph of each
function.
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2Solve: log 2 1 3 log 343 3xa x b
2(a) Change log 2 1 3 to exponential form.x
32 2 1x 8 2 1x 7
2x
(b) Change log 343 3 to exponential form.x
3 343x 7x
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a.
b.
c.
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MATH 108
Section 4.4
Properties of Logarithms
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RULES OF LOGARITHMSLet M, N, and a be positive real numbers with a ≠ 1, and let r be any real
number.
The logarithm of the product of two (or more) numbers is the sum of the
logarithms of the numbers.
1. Product Rule
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RULES OF LOGARITHMSLet M, N, and a be positive real numbers with a ≠ 1, and let r be any real
number.
The logarithm of the quotient of two (or more) numbers is the difference of the
logarithms of the numbers.
2. Quotient Rule
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RULES OF LOGARITHMSLet M, N, and a be positive real numbers with a ≠ 1, and let r be any real
number.
The logarithm of a number to the power r is r times the logarithm of the
number.
3. Power Rule
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2 32Write log 1 , >1, as a sum of logarithms.
Express all powers as factors.
x x x
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6 22Write log , 0, as a difference of logarithms.
3
Express all powers as factors.
xx
x
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3
2
2Write ln , 2, as a sum and difference of logarithms.
1Express all powers as factors.
x xx
x
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EXAMPLE 2 Writing Expressions In Expanded Form
32
2 4
1a. log
2 1
x x
x
Write each expression in expanded form.
3 2 5b. ln x y z
Solution
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2
Write each of the following as a single logarithm.
a 3ln 2 ln
1b log 4 2log 5
2
c 2log 3 3log 2 log 1
a a
a a a
x
x
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EXAMPLE 3 Writing Expressions in Condensed Form
a. log3 log 4x y
Write each expression in condensed form.
21b. 2ln ln 1
2x x
2 2 2c. 2log 5 log 9 log 75
21d. ln ln 1 ln 1
3x x x