A Double Exponential Equation

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Since and , there are points on the graphs of and where . These graphs are the special cases of where and . All points with can be found as intersections of the graph with the lines with slope . In this case, parametric equations in terms of have simple formulas.

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The graph of is black. The graph of interest, where , is blue for and red for , and is the graph of a function . The intersection points with and , for , and corresponding points on , are plotted.

It is interesting to see that when is varied between 0 and 2, the graph of bows from concave up to concave down, and appears to be a line segment from to for some . The graphs of and are shown to help you decide whether the graph of for this really is straight. The special satisfies .

The case is especially interesting because then the equation is equivalent to , which has a solution and . (The slider for can take values from -2 to 5.)

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Contributed by: Roger B. Kirchner (March 2011)
Open content licensed under CC BY-NC-SA


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Details

Assume . Conveniently, iff iff iff . Thus, is on the graph of where iff and , where .

Let and . Then and , and are parametric equations for the graph of . Since the graph of is symmetric with respect to , and .

Parametric equations for the graphs of and are obtained by differentiating the identity . They are , , and , .

Where does the graph of meet the line ? Two ways to determine it are 1) observe it is the point such that , where , and 2) compute using L'Hospital's rule. You will find .

What are the domain and range of ? It is an interesting exercise in L'Hospital's rule to determine that and when and and when . By symmetry, the domain and range of are when and when .



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