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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 56
Chapter 5, Problem 56

Begin by graphing f(x) = log₂ x. Then use transformations of this graph to graph the given function. What is the vertical asymptote? Use the graphs to determine each function's domain and range. h(x) = 2 + log2x

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Start by understanding the base function: \(f(x) = \log_{2} x\). This function has a vertical asymptote at \(x = 0\), a domain of \((0, \infty)\), and a range of \((-\infty, \infty)\).
Next, analyze the given function \(h(x) = 2 + \log_{2} x\). Notice that this is a vertical shift of the base function \(f(x)\) upward by 2 units.
Since adding 2 shifts the graph vertically, the vertical asymptote remains unchanged at \(x = 0\) because vertical asymptotes depend on the input values where the function is undefined, which is not affected by vertical shifts.
Determine the domain of \(h(x)\) by considering where \(\log_{2} x\) is defined. Since \(\log_{2} x\) is defined for \(x > 0\), the domain of \(h(x)\) is also \((0, \infty)\).
Determine the range of \(h(x)\) by shifting the range of \(f(x)\) up by 2. Since the range of \(f(x)\) is \((-\infty, \infty)\), the range of \(h(x)\) remains \((-\infty, \infty)\) because adding a constant shifts the entire range but does not restrict it.

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Key Concepts

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

Graphing Logarithmic Functions

Graphing logarithmic functions involves plotting points based on the logarithm's definition and shape. For f(x) = log₂ x, the graph passes through (1,0), increases slowly, and has a vertical asymptote at x = 0. Understanding this base graph is essential before applying transformations.
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Graphs of Logarithmic Functions

Transformations of Functions

Transformations modify the graph of a base function by shifting, stretching, or reflecting it. For h(x) = 2 + log₂ x, the '+2' shifts the graph vertically upward by 2 units, affecting the range but not the vertical asymptote or domain.
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Domain & Range of Transformed Functions

Domain, Range, and Vertical Asymptotes of Logarithmic Functions

The domain of log₂ x is (0, ∞) because logarithms are undefined for non-positive values. The vertical asymptote is the line x = 0, where the function approaches negative infinity. Adding a constant shifts the range but does not change the domain or asymptote.
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Determining Vertical Asymptotes
Related Practice
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+4)=−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. 5 ln x - 2 ln y

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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. (1/2)ln x - ln y

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Textbook Question

Graph f and g in the same rectangular coordinate system. Then find the point of intersection of the two graphs. f(x) = 2x, g(x) = 2-x

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. log2(x+25)=4

Textbook Question

Use the compound interest formulas A = P (1+ r/n)nt and A =Pert to solve exercises 53-56. Round answers to the nearest cent. Suppose that you have \$12,000 to invest. Which investment yields the greater return over 3 years: 0.96% compounded monthly or 0.95% compounded continuously?