# 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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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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