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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 59
Chapter 5, Problem 59

Graph y= 2x and x = 2y in the same rectangular coordinate system.

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Identify the two functions to be graphed: the first is an exponential function \(y = 2^{x}\), and the second is \(x = 2^{y}\), which can be rewritten to express \(y\) in terms of \(x\) for easier graphing.
Rewrite the second equation \(x = 2^{y}\) by taking the logarithm base 2 of both sides to solve for \(y\): \(y = \log_{2}(x)\). This shows that the second graph is the logarithmic function, the inverse of the first.
Plot key points for \(y = 2^{x}\) by choosing several values of \(x\) (such as \(-2, -1, 0, 1, 2\)) and calculating the corresponding \(y\) values using \(y = 2^{x}\).
Plot key points for \(y = \log_{2}(x)\) by choosing several positive values of \(x\) (such as \(\frac{1}{4}, \frac{1}{2}, 1, 2, 4\)) and calculating the corresponding \(y\) values using \(y = \log_{2}(x)\).
Draw smooth curves through the plotted points for both functions on the same coordinate system, noting that the graphs are reflections of each other across the line \(y = x\) because they are inverse functions.

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Key Concepts

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

Exponential Functions

An exponential function has the form y = a^x, where the base a is a positive constant not equal to 1. It models rapid growth or decay, and its graph passes through (0,1) with a smooth curve increasing or decreasing depending on the base. Understanding y = 2^x helps in plotting one of the given functions.
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Exponential Functions

Inverse Functions and Their Graphs

The function x = 2^y can be seen as the inverse of y = 2^x by swapping variables. Inverse functions reflect across the line y = x, so graphing both on the same axes shows this symmetry. Recognizing this relationship aids in understanding the shape and position of both graphs.
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Graphs of Logarithmic Functions

Coordinate System and Plotting Points

Plotting functions requires understanding the rectangular coordinate system, where each point is defined by (x, y). To graph y = 2^x and x = 2^y, select values for x or y, compute corresponding points, and plot them accurately. This visual representation helps compare and analyze the functions.
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Graphs & the Rectangular Coordinate System
Related Practice
Textbook Question

In Exercises 53-58, 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. g(x) = (1/2)log₂ x

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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

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. 4 ln (x + 6) - 3 ln x

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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

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Textbook Question

In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log4 0.863

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Textbook Question

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. g(x) = log(x − 1)

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