Find the number $n^{2}$ from the number $n^{n^{n^{2}}}$

Question

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$

Any help? I tried with $\log$ but I got nothing.

Answer 1

A slight extension of the idea to take logs suggested by OP and @CameronWilliams can be used to get an inelegant partial solution based on the obvious computations. While it does go farther than I thought possible from the Comment by @EricTressler, one can hope that a general solution can be found in closed form that does not depend on computation.

For $n = 2, \dots 25,\;$ let $q = n^2,\; h = n^{n^q}$ and $$c = \lfloor \log_{10}(\log_{10}(h))\rfloor = \lfloor q \log_{10}(n) + \log_{10}(\log_{10}(n))\rfloor.$$ Use $c$ in the following table to find $n$ and $q.\;$ (If $h = 1,$ then $n = n^2 = 1.$)

      n   q   c
      ---------
      2   4   0
      3   9   3
      4  16   9
      5  25  17
      6  36  27
      7  49  41
      8  64  57
      9  81  77
     10 100 100
     11 121 126
     12 144 155
     13 169 188
     14 196 224
     15 225 264
     16 256 308
     17 289 355
     18 324 406
     19 361 461
     20 400 520
     21 441 583
     22 484 649
     23 529 720
     24 576 795
     25 625 873

With suitable software this table might be extended to somewhat larger values of $n,$ but this table illustrates the idea. Maybe it will give someone a clue towards a closed form solution.