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Chapter 3: Problem 43

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

Short Answer

Expert verified
Both \(f(x)=4^{x}\) and \(g(x)=\log_{4} x\) are graphed on the same coordinate system. The graph of the exponential function passes through the point (0,1) and increases as it moves to the right. The graph of the logarithmic function passes through the point (1,0) and increases as it moves to the right, but at a decreasing rate. They are mirror images about the line \(y=x\).

Step by step solution

01

Graph of exponential function \(f(x)=4^{x}\)

First, let's deal with \(f(x)=4^{x}\). The exponential function \(f(x) = 4^{x}\) passes through the point (0,1) and has a horizontal asymptote, the x-axis or \(y=0\). When \(x=1\), \(y=4\), and as \(x\) becomes increasingly negative, \(y\) approaches 0 but never quite reaches it. Plot these values to get an increasing curve from left to right.
02

Graph of logarithmic function \(g(x)=\log _{4} x\)

Then look at \(g(x)=\log_{4} x\). Logarithmic functions are the inverse of exponential functions, so the graph of \(g(x)=\log_{4} x\) can be obtained by reflecting the graph of \(f(x)=4^{x}\) in the line \(y=x\). This function is not defined for \(x \leq 0\). Plot a point at (1,0) and another point at (4,1), as these are easy to calculate and are part of the function. The function increases as x increases, but at a decreasing rate.
03

Combine graphs of \(f(x)=4^{x}\) and \(g(x)=\log_{4} x\)

Now plot \(f(x)=4^{x}\) and \(g(x)=\log_{4} x\) on the same graph. They should be mirror images of each other about the line \(y = x\), with one graph covering the first and fourth quadrants (the exponential function) and the other covering the first and second quadrants (the logarithmic function).

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Most popular questions from this chapter

Exercises \(153-155\) will help you prepare for the material covered in the next section. a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)

Graph \(f(x)=2^{x}\) and its inverse function in the same rectangular coordinate system.

Graph \(f\) and \(g\) in the same rectangular coordinate system. Then find the point of intersection of the two graphs. Graph \(y=2^{x}\) and \(x=2^{y}\) in the same rectangular coordinate system.

$$\text { Solve for } y: 7 x+3 y=18$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\).
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