For very large calculations Wolfram Alpha offers a variety of different representations of the number. One of these is the number written in the form $10^{10^n}$, where $n$ is usually some long decimal, such as in this example .
Does any one know how this is done? I cannot image the calculation of the entire number is done and then just converted to that form but I could not think of any other way it could be done.
If you're equating $x=10^{10^n}$, then $n=\log(\log(x))$ (where "$\log$" is understood to mean base $10$). If $x=5515896^{678435}$, then $$\log x=678435\log(5515896)\approx4573748.295$$ hence $$n=\log(\log(x))=\log(678345\log(5515896))\approx\log(4573748.295)\approx6.66027$$