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Chapter 12: Problem 44

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

Short Answer

Expert verified
Both the functions f and g are plotted on the same rectangular coordinate system. They intersect at x=1, and each function mirrors the other across the line y=x.

Step by step solution

01

1. Plotting the Exponential Function

The exponential function \(f(x) = 5^{x}\) reflects the characteristic that the y-value increases rapidly with the increase of x. When x=0, y=1. By choosing a set of values for x and finding their corresponding y-values, we get a set of coordinates that can be plotted. Label the axes of the graph properly.
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2. Plotting the Logarithmic Function

The function \(g(x) = \log_{5}{x}\) reflects that the y-value rises with the increase of x, but at a much slower rate than the exponential function. This function is not defined for x less than or equal to 0. By choosing a set of values for x and finding their corresponding y-values, we get a another set of coordinates that can be plotted on the same graph. Label it properly.
03

3. Analyze the Graphs

Now both functions are plotted on the same graph, it is visually clearly that they intersect at x=1. Also, we can observe that these functions mirror each other across the line y=x, which becomes apparent since the exponential function and the logarithmic function are inverses of each other.

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Most popular questions from this chapter

Explain how to find the domain of a logarithmic function.

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Will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{2} 32\) b. Evaluate: \(\log _{2} 8+\log _{2} 4\) c. What can you conclude about \(\log _{2} 32,\) or \(\log _{2}(8 \cdot 4) ?\)

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