How to compare large exponents with different bases? Is there any way to roughly approximate their values?

For example, sort the elements of list below based on their magnitude.

$381600^{809197}, 105964^{708702}, 149040^{415447}, 289337^{847908}, 789760^{296736}$

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    $\begingroup$ Taking logarithms? Maybe double logs: $\log(\log(a^b))=\log(b)+\log(\log(a))$. They are monotonic functions, so you can compare the logs. $\endgroup$
    – user115760
    Commented Aug 16, 2015 at 11:09
  • 1
    $\begingroup$ Language nitpick: The exponents in your list are 809197, 708702, 415447, 847908, and 296736. These are easy to compare. What you want to compare is the powers. $\endgroup$ Commented Aug 16, 2015 at 11:20
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    $\begingroup$ But actually it's enough in most cases to compare the exponents -- the logarithms of the bases are all between 5 and 6, so the much wider differences in the exponents are going to dominate. The only comparison that seems to be in risk of not being determined by the exponents is $381600^{809197}$ versus $289337^{847908}$. $\endgroup$ Commented Aug 16, 2015 at 11:25
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    $\begingroup$ It would be very laborious to factorize these numbers and then compare them. It might be faster and more reliable to put them through Wolfram Alpha. For example, WA tells me that $381600^{809197} \approx 5.5 \times 10^{4516620}$. $\endgroup$ Commented Aug 16, 2015 at 18:16


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