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Chapter 12: Problem 51

Graph \(y=2^{x}\) and \(x=2^{y}\) in the same rectangular coordinate system.

Short Answer

Expert verified
After graphing the two functions, \(y=2^{x}\) appears as a curve passing through (0,1) that increases as x increases. The graph of \(x=2^{y}\) (or \(y=\log_{2}{x}\)) starts from the y-axis and increases as y increases. Both graphs intersect at two points: (0,1) and (1,2).

Step by step solution

01

- Graph the function \(y=2^{x}\)

To graph the function \(y=2^{x}\), we start by creating a table of values then plot these values on the rectangular coordinate system. The exponential function \(y=2^{x}\) will result in a graph that is always above the x-axis, passes through the point (0,1) and increases as x increases.
02

- Graph the function \(x=2^{y}\)

To graph the function \(x=2^{y}\), we first rewrite it as \(y=\log_{2}{x}\), by applying the logarithm definition. Then similar to the previous step, we create a table of values and then plot these values in the same rectangular coordinate system. The resulting graph starts from the y-axis and increases as y increases.
03

- Identify Intersections

By comparing the two functions visually on the graph, we can see where they intersect. These intersections provide the solutions to the equation \(2^{x} = \log_{2}{x}\).

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Most popular questions from this chapter

Explain how to find the domain of a logarithmic function.

Explain why the logarithm of 1 with base \(b\) is \(0 .\)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\ln \sqrt{2}=\frac{\ln 2}{2}$$

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the \(\operatorname{graph}\) of \(g(x)=\log _{2} x\)

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. \(\log _{3}(3 x-2)=2\)
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