Find all values of b such that the equation
b^{x} = log_{b}x
has exactly one real solution.
(In reply to
Solution by Harry)
As broll has noted, when b < e^(e) ~= 0.06598803585 there are two solutions. As an example, when b~= 0.05 there's a second solution where x ~= .1373594001 and y ~= 0.6626608316, in addition to the solution where x=y (x ~= 0.35022485364, y = x).
Both curves are decreasing and concave up, one (the log curve) having more concavity than the other allowing for the two solutions.

Posted by Charlie
on 20121213 16:18:18 