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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 61
Chapter 5, Problem 61

The figure shows the graph of f(x) = log x. In Exercises 59–64, 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) = log x − 1
Graph of f(x) = log x showing points (1,0), (5,0.7), (10,1) and vertical asymptote at x = 0.

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Start with the base function given: \(f(x) = \log x\). Recall that the graph of \(\log x\) has a vertical asymptote at \(x = 0\), a domain of \((0, \infty)\), and a range of \((-\infty, \infty)\).
Identify the transformation in the function \(h(x) = \log x - 1\). This represents a vertical shift downward by 1 unit of the original graph \(f(x) = \log x\).
Apply the vertical shift to the graph: move every point on the graph of \(f(x) = \log x\) down by 1 unit. This means the shape of the graph remains the same, but the entire curve is lowered by 1.
Determine the new equation of the asymptote. Since vertical shifts do not affect vertical asymptotes, the vertical asymptote remains at \(x = 0\).
Find the domain and range of \(h(x)\). The domain remains \((0, \infty)\) because the logarithm's input hasn't changed. The range shifts down by 1, so the new range is \((-\infty, \infty)\) shifted down by 1, which is still \((-\infty, \infty)\).

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Key Concepts

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

Logarithmic Functions

A logarithmic function, such as f(x) = log x, is the inverse of an exponential function. It is defined only for positive x-values and has a vertical asymptote at x = 0. Understanding its basic shape and properties is essential for graphing transformations.
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Graphs of Logarithmic Functions

Transformations of Functions

Transformations involve shifting, stretching, or reflecting the graph of a function. For h(x) = log x − 1, subtracting 1 shifts the graph downward by 1 unit, affecting the range but not the domain or vertical asymptote.
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Domain & Range of Transformed Functions

Asymptotes, Domain, and Range

The vertical asymptote of log functions occurs where the argument is zero (x=0). The domain is the set of x-values where the function is defined (x > 0), and the range is all real numbers, which may shift with transformations.
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Determining Horizontal Asymptotes
Related Practice
Textbook Question

In Exercises 60–63, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. (log2 x)^4 = 4 log2 x

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Textbook Question

Give the equation of each exponential function whose graph is shown.

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. 6+2 ln x=5

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. (1/2)(log x + log y)

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. 3 ln x + 5 ln y - 6 ln z

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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. 5 ln(2x)=20