Textbook Question
In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log5 13
Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 5, Problem 69
The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = 2 ln x
Verified step by step guidance1
Recall the parent function: \(f(x) = \ln x\), which has a vertical asymptote at \(x = 0\), domain \((0, \infty)\), and range \((-\infty, \infty)\).
Identify the transformation in \(g(x) = 2 \ln x\): the factor 2 is a vertical stretch by a factor of 2, which stretches the graph of \(\ln x\) vertically but does not affect the domain or the asymptote.
Write the equation of the asymptote for \(g(x)\): since the transformation does not shift the graph horizontally, the vertical asymptote remains at \(x = 0\).
Determine the domain of \(g(x)\): because the input to the logarithm must be positive, the domain remains \((0, \infty)\).
Determine the range of \(g(x)\): since vertical stretching does not restrict the output values of the logarithm, the range remains \((-\infty, \infty)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Functions and Their Graphs
A logarithmic function, such as f(x) = ln x, is the inverse of an exponential function. Its graph passes through (1,0) and is defined only for x > 0. Understanding the shape and behavior of the natural logarithm graph is essential for applying transformations.
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Graphs of Logarithmic Functions
Transformations of Functions
Transformations include vertical stretches, compressions, shifts, and reflections applied to a base graph. For g(x) = 2 ln x, the factor 2 vertically stretches the graph of ln x by a factor of 2, affecting the steepness but not the domain or asymptote location.
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4:22Domain & Range of Transformed Functions
Asymptotes, Domain, and Range of Logarithmic Functions
The vertical asymptote of ln x is the y-axis (x=0), where the function is undefined. The domain of ln x and its transformations is x > 0, while the range is all real numbers. Recognizing how transformations affect these properties is key to graphing and describing the function.
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5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
In Exercises 64–73, solve each exponential equation. Where necessary, express the solution set in terms of natural or common logarithms and use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.
Textbook Question
The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range.
h(x) = ln (2x)
Textbook Question
Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log3(x+6)+log3(x+4)=1
Textbook Question
The figure shows the graph of f(x) = ln x. In Exercises 65–74, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = ln(x/2)
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.