171S5.3_p Logarithmic Functions
MAT 171 Precalculus Algebra
Trigsted ­ Pilot Test
Dr. Claude Moore ­ Cape Fear Community College
CHAPTER 5: Exponential and Logarithmic Functions and Equations
5.1 Exponential Functions
5.2 The Natural Exponential Function
5.3 Logarithmic Functions
5.4 Properties of Logarithms
5.5 Exponential and Logarithmic Equations 5.6 Applications of Exponential and Logarithmic Functions
Logarithmic Functions
For x > 0, b > 0, and b ≠ 1
The logarithmic function with base b is defined by
y = logb x if and only if x = by
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Objectives
1. Understanding the Definition of a Logarithmic Function
2. Evaluating Logarithmic Expressions
3. Understanding the Properties of Logarithms
4. Using the Common and Natural Logarithms
5. Understanding the Characteristics of Logarithmic Functions
6. Sketching the Graphs of Logarithmic Functions Using
Transformations
7. Finding the Domain of Logarithmic Functions.
Write each exponential equation as
an equation involving a logarithm.
y = logb x
if and only if x = by
If f(x) = b x, then f ­1(x) = logb x.
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171S5.3_p Logarithmic Functions
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Write each logarithmic equation as
an equation involving an exponent.
y = logb x
if and only if x = by
Evaluate each logarithm
y = logb x if and only if x = by
Properties of Logarithms
For b > 0 and b ≠ 1
1. logb b = 1
2. logb 1 = 0
3. blogb x = x
4. logb bx = x
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171S5.3_p Logarithmic Functions
Use the properties of logarithms to evaluate each expression.
y = logb x iff x = by
2.1
2.1
y
y = log17 17 iff 17 = 17
y = log6.1 6.1 iff 6.1 = 6.1y
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Common Logarithmic Functions (base 10)
For x > 0, the common logarithmic function is defined
by y = log x if and only if x = 10y
Natural Logarithmic Functions (base e)
y
y = log532 1 iff 1 = 532
x = 9log9 43 iff log9 43 = log9 x
Write each exponential equation as an equation involving a common or natural logarithm.
For x > 0, the natural logarithmic function is defined
by y = ln x if and only if x = ey
Evaluate each expression without the use of a calculator.
Write each logarithmic equation as an equation involving an exponent.
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171S5.3_p Logarithmic Functions
How to sketch the graph of a logarithmic function of the form f(x) = logb x, where b > 0, and b ≠ 1
Step 1: Start with the graph of the exponential function y = bx labeling several ordered pairs. April 14, 2011
Step 2: Because the logarithmic and exponential functions are inverses, several points on the logarithmic function can be found by reversing the coordinates of the exponential function. Step 3: Connect the ordered pairs with a smooth curve. Characteristics of Logarithmic Functions
For x > 0, b > 0, b ≠ 1, the logarithmic function with base b has a domain of (0, ∞) and a range of (­∞, ∞). The graph has one of the following two shapes depending on the value of b. f(x) = logb x, b > 1
Graph intersects the x­axis at (1,0).
Graph contains the point (b,1).
Graph increases on the interval (0, ∞). y­axis (x = 0) is a vertical asymptote.
The function is one­to­one.
f(x) = logb x, 0 < b < 1
Graph intersects the x­axis at (1,0).
Graph contains the point (b,1).
Graph decreases on the interval (0, ∞).
y­axis (x = 0) is a vertical asymptote.
The function is one­to­one.
Sketch the graph of f(x) = ­ln(x­3) +2 using transformations.
1. Begin with the graph of f(x) = ln x.
2. Shift the graph 3 units right f(x) = ln(x – 3).
3. Reflect the graph about the x axis
f(x) = ­ln(x – 3)
4. Shift the graph 2 units up
f(x) = ­ln(x – 3) + 2
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Domain of a logarithmic function consists of all values of x for which the argument of the logarithm is greater than zero. In other words, if y = logb x iff x = by
f(x) = logb [g(x)],
then the domain of f(x) can be found by solving the inequality g(x) > 0.
For the function f(x) = ­ ln(x ­ 3) + 2, the domain is found by solving
x ­ 3 > 0 to get x > 3.
The domain is (3, ∞).
y = logb x iff x = by
y = logb x iff x = by
If bx = by, then x = y.
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171S5.3_p Logarithmic Functions
For b > 0 and b ≠ 1
1. logb b = 1
2. logb 1 = 0
3. blogb x = x
4. logb bx = x
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y = logb x iff x = by
y = logb x iff x = by
For b > 0 and b ≠ 1
1. logb b = 1
2. logb 1 = 0
3. blogb x = x
4. logb bx = x
y = logb x iff x = by
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