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