Solution to Problem 265

Correct solutions came from Vincent Lynch, UK; Juan Carlos Marivela, Espaņa; Brett Hendricks, USA; Aaron Kahn; Philippe Fondanaiche, France; Jack Tseng; Bill Webb, USA; Ron Welch, USA; John Snyder, USA, Jordan Garsson, USA.

The equation ax = x2 is algebraically equivalent to ln(a)/2 = ln(x)/x.  The graph of the left-hand side is a horizontal line with y-intercept ln(a)/2, that of the right-hand side is shown on the left.  The first equation has only one solution exactly when the second does.  This occurs at the maximum value of x = e in which case a = e2/e, or if a ≤ 1.  

(Thanks to fifth-grader Jordan Garsson for a correction to my original solution.  Way to go Jordan!)

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