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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 40
Chapter 5, Problem 40

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = ln x and g(x) = - ln (2x)
Graph of f(x) = ln x (red) and g(x) = -ln(2x) (blue) with asymptotes and key points.
Graph of g(x) = -ln(2x) (red) and f(x) = ln x (blue) with transformations and key points.
Graph of f(x) = ln x (red) and g(x) = -ln(2x) (blue) with asymptotes and key points.
Graph of g(x) = -ln(2x) (red) and f(x) = ln x (blue) with transformations and key points.

Verified step by step guidance
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Step 1: Begin by analyzing the given functions f(x) = ln(x) and g(x) = -ln(2x). The function f(x) = ln(x) is the natural logarithm function, which has a vertical asymptote at x = 0 and is defined for x > 0. The function g(x) = -ln(2x) involves a reflection across the x-axis and a horizontal compression by a factor of 2.
Step 2: Identify the transformations applied to f(x) to obtain g(x). The negative sign in g(x) = -ln(2x) reflects the graph of f(x) across the x-axis, and the factor of 2 inside the logarithm compresses the graph horizontally. This means that for g(x), the x-values are scaled by a factor of 1/2 compared to f(x).
Step 3: Determine the asymptotes for both functions. For f(x) = ln(x), the vertical asymptote is at x = 0 because the natural logarithm is undefined for x ≤ 0. Similarly, g(x) = -ln(2x) also has a vertical asymptote at x = 0, as the argument of the logarithm (2x) must be positive.
Step 4: Use the graphs to determine the domain and range of each function. For f(x) = ln(x), the domain is (0, ∞) and the range is (-∞, ∞). For g(x) = -ln(2x), the domain is also (0, ∞), but the range is (-∞, ∞) due to the reflection across the x-axis.
Step 5: Plot key points to verify the transformations. For f(x), key points include (1, 0) and (e, 1). For g(x), the corresponding points after transformation are (0.5, 0) and (e/2, -1). These points confirm the horizontal compression and reflection applied to f(x) to obtain g(x).

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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) = ln(x), are the inverses of exponential functions. They are defined for positive real numbers and have a vertical asymptote at x = 0. Understanding their properties, including their domain, range, and behavior as x approaches the asymptote, is crucial for graphing and analyzing transformations.
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Graphs of Logarithmic Functions

Transformations of Functions

Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, g(x) = -ln(2x) represents a vertical reflection of f(x) = ln(x) and a horizontal compression. Recognizing how these transformations affect the graph helps in visualizing and deriving the new function's characteristics from the original.
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Domain & Range of Transformed Functions

Asymptotes

Asymptotes are lines that a graph approaches but never touches. For logarithmic functions, vertical asymptotes occur where the function is undefined, such as at x = 0 for f(x) = ln(x). Identifying asymptotes is essential for understanding the behavior of the function at its boundaries and for determining the domain and range of the functions involved.
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Introduction to Asymptotes
Related Practice
Textbook Question

In Exercises 39–40, graph f and g in the same rectangular coordinate system. Use transformations of the graph of f to obtain the graph of g. Graph and give equations of all asymptotes. Use the graphs to determine each function's domain and range. f(x) = log x and g(x) = - log (x+3)

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Textbook Question

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log 5 + log 2

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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. log(10x21−x37(x+1)2)\(\log\) \(\left\)( \(\frac{10x^2 \sqrt[3]{1 - x}\)}{7(x + 1)^2} \(\right\))

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Textbook Question

Evaluate each expression without using a calculator. log5 57

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Textbook Question

Evaluate each expression without using a calculator. 8log8198^{\(\log\)_8 19}

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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. 5(2x+3)=3(x−1)

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