Correct solutions were sent in by Philippe Fondanaiche,
France; Bill Webb, USA; Farid Lian, Colombia; Ron Welch, USA.
First take the natural logarithm of both sides and
rearrange the equation to read x/ln(x) = y/ln(y). Let w = f (u) = u/ln(u); we are looking
for points where the function f
takes on the same value more than once. A little
calculus will verify that this occurs for all values of w > e.
Now, the algebra:
Notice that in going from the sixth to to the seventh
equation, we must exclude y
≠ x;
and, in fact, x
= y is an
obvious solution to the equation. Let a = y/x ≠ 1, then
is a parametric
solution to the equation.
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