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Chapter 3: Problem 44

Graph \(f(x)=5^{x}\) and \(g(x)=\log _{5} x\) in the same rectangular coordinate system.

Short Answer

Expert verified
The graph of \(f(x) = 5^x\) passes through the points (-2, 0.04), (-1, 0.2), (0,1), (1, 5) and (2, 25). The graph of \(g(x) = \log_{5}x\) passes through the points (1, 0), (5, 1), (25, 2). The functions increase in the positive direction.

Step by step solution

01

Plotting the Exponential Function

Plotting the function \(f(x) = 5^x\), take a few values for x (-2, -1, 0, 1, 2) and calculate the corresponding y values which are: \(f(-2) = 5^{-2}= 0.04\), \(f(-1) = 5^{-1}=0.2\), \(f(0) = 5^0=1\), \(f(1) = 5^{1}=5\), and \(f(2) = 5^{2}=25\). Mark these points on the rectangular coordinate system.
02

Plotting the Logarithmic Function

Plotting the function \(g(x) = log_{5}x\), take a few values for x (1, 5, 25) and calculate the corresponding y values which are: \(g(1) = log_{5}1=0\), \(g(5) = log_{5}5=1\), and \(g(25) = log_{5}25=2\). Remember that logarithmic functions are not defined for \(x \leq 0\), so plot these points on the same rectangular coordinate system.
03

Drawing the graphs

Use the points identified to draw the curves of each function. The exponential function \(5^x\) is increasing and passes through points where \(x=-2, -1, 0, 1, 2\). The logarithmic function \(\log_{5}x\) is also increasing and passes through points where \(x=1, 5, 25\). Remember that the logarithmic function isn't defined for negative x values.

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Most popular questions from this chapter

Solve each equation in Exercises \(146-148 .\) Check each proposed solution by direct substitution or with a graphing utility. $$(\log x)(2 \log x+1)=6$$

Graph each of the following functions in the same viewing rectangle and then place the functions in order from the one that increases most slowly to the one that increases most rapidly. $$y=x, y=\sqrt{x}, y=e^{x}, y=\ln x, y=x^{x}, y=x^{2}$$

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

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

In Exercises \(141-144,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Examples of exponential equations include \(10^{x}=5.71\) \(e^{x}=0.72,\) and \(x^{10}=5.71\)
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