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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 36
Chapter 5, Problem 36

In Exercises 36–38, begin by graphing f(x) = log2 x Then use transformations of this graph to graph the given function. What is the graph's x-intercept? What is the vertical asymptote? Use the graphs to determine each function's domain and range. g(x) = log2 (x-2)

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Step 1: Start by understanding the base function f(x) = log₂(x). This is the logarithmic function with base 2. Its graph passes through the point (1, 0), has a vertical asymptote at x = 0, and is defined for x > 0. The domain of f(x) is (0, ∞), and the range is (-∞, ∞).
Step 2: Analyze the transformation in g(x) = log₂(x - 2). The term (x - 2) indicates a horizontal shift of the graph of f(x) to the right by 2 units. This means every point on the graph of f(x) will move 2 units to the right.
Step 3: Apply the horizontal shift to the key features of the graph of f(x). The vertical asymptote, originally at x = 0, will now shift to x = 2. The x-intercept, originally at (1, 0), will shift to (3, 0).
Step 4: Determine the domain and range of g(x). Since the graph is shifted to the right by 2 units, the domain of g(x) is (2, ∞). The range remains unchanged as (-∞, ∞), because logarithmic functions are not affected vertically by horizontal shifts.
Step 5: Summarize the key features of g(x). The x-intercept is at (3, 0), the vertical asymptote is at x = 2, the domain is (2, ∞), and the range is (-∞, ∞). Use these features to sketch the graph of g(x), ensuring it reflects the horizontal shift of the base function.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithmic Functions

Logarithmic functions, such as f(x) = log2 x, are the inverses of exponential functions. They are defined for positive real numbers and have a characteristic shape that approaches the vertical axis (y-axis) but never touches it, indicating a vertical asymptote at x = 0. Understanding the properties of logarithmic functions is essential for analyzing their graphs and transformations.
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Graphs of Logarithmic Functions

Transformations of Functions

Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, g(x) = log2 (x-2) represents a horizontal shift of the graph of f(x) = log2 x to the right by 2 units. Recognizing how these transformations affect the graph is crucial for determining features like intercepts, asymptotes, and the overall shape of the function.
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Domain & Range of Transformed Functions

Domain and Range

The domain of a function refers to all possible input values (x-values) for which the function is defined, while the range refers to all possible output values (y-values). For logarithmic functions, the domain is typically restricted to values greater than zero, and the range is all real numbers. Understanding the domain and range helps in identifying the behavior of the function and its graph.
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Domain & Range of Transformed Functions
Related Practice
Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 7(x+2)=410

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Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5x2y243\(\log\)_5 \(\sqrt\)[3]{\(\frac{x^2 y}{24}\)}

Textbook Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(x3x2+1(x+1)4)\(\ln\) \(\left\)( \(\frac{x^3 \sqrt{x^2 + 1}\)}{(x + 1)^4} \(\right\))

Textbook Question

Evaluate each expression without using a calculator. log5 5

Textbook Question

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = ex+2

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Textbook Question

The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn g(x) = ex-1