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Chapter 4: Problem 39

Graph \(f(x)=4^{x}\) and \(g(x)=\log _{4} x\) in the same rectangular coordinate system.

Short Answer

Expert verified
The important relationship to note from the graphing of these two functions comes from the properties of logarithmic and exponential functions. As seen in the graphical representation, the two graphs are mirror images of each other along the line \(y=x\). This is due to the fact that logarithmic and exponential functions are inverses of each other.

Step by step solution

01

Graph the exponential function

Start by graphing the function \(f(x)=4^{x}\). Remember that any exponential function will always pass through the point (0,1) because anything raised to the power of 0 is 1. Also, because the base 4 is greater than 1, the function will be increasing. Thus, to the right, the graph rises, and to the left, it gets closer and closer to the x-axis but never touches or crosses it.
02

Graph the logarithmic function

Now, graph the logarithmic function \(g(x)=\log _{4} x\). The point (4,1) will always be on the logarithmic curve because \(\log_4 4 = 1\). Also, since logarithms are undefined for zero and negative numbers, the graph will only exist to the right of the y-axis. The graph of \(g(x)=\log _{4} x\) will resemble somewhat of a mirror image of the exponential function along the line \(y=x\), thanks to the logarithmic and exponential functions being inverses of each other.
03

Overlay both graphs onto a single coordinate system

Upon completing the graphs of both functions, overlay both onto the same rectangular coordinate system. This will allow for a direct comparison between the functions and a clear visual representation of their properties and relationships, notably that they are mirror images of each other across the line \(y=x\) in the coordinate.

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Most popular questions from this chapter

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{8}\left(\frac{64}{\sqrt{x+1}}\right) $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt{e x} $$

The loudness level of a sound can be expressed by comparing the sound's intensity to the intensity of a sound barely audible to the human car. The formula $$ D=10\left(\log I-\log I_{0}\right) $$ describes the loudness level of a sound, \(D\), in decibels, where \(I\) is the intensity of the sound, in watts per meter". and \(I_{0}\) is the intensity of a sound barely audible to the human ear. a. Express the formula so that the expression in parentheses is written as a single logarithm. b. Use the form of the formula from part (a) to answer this question: If a sound has an intensity 100 times the intensity of a softer sound, how much larger on the decibel scale is the loudness level of the more intense sound?

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log (2 x+5)-\log x $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{9}\left(\frac{9}{x}\right) $$
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