Exactly this example is treated as a prototype for a broad family of
similar problems in a paper by Szekeres; he argues that there is one
particularly optimal solution  y and in an accompanying paper provides
tables of numerical values. (He uses the analysis to describe a
one-parameter family of functions  f_s  with  f_s o f_t = f_{s+t} and
in particular describes a functional "square root"  f_{1/2}  of  f = f_1.)

   MR0141905 (25 #5302)
   Szekeres, G.
   Fractional iteration of exponentially growing functions.
   J. Austral. Math. Soc. 2 1961/1962 301--320.        MSC section 39.99