The point I was getting at is that 2**1024000, is already logarithmic. Ie. 1024000 is log₂ of your number.

Thus, to convert it to log₁₀(), you only need divide it by log₂( 10 ).

Then all you need is:

sub log2{ log( $_[0] ) / log( 2 ) }
sub log2ToLog10{ $_[0] / log2( 10 ) }

print log2ToLog10( 1024000 );;
308254.715559917
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And once you have that, displaying it in human readable form becomes:

my $log10 = log2ToLog10( 1024000 );

printf "%.16fe%d\n",  10**( $log10 - int( $log10 )), int( $log10 );;
5.1946933632179482e308254
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Which is pretty accurate according to wolfram/alpha:

5.19469336319992513461071316398707637029610983547139039022052215500033
+0748128713140857944929474604946407890576242141198731139214
2324060603669100591443085161671423608538064065009247433495758206421181
+6041163544936451093418528960464769157767991031321856057398
2666947442696473760475992972158927479218009273759516049811496993843563
+6416559392963943454835743787169456588299548100913424497325
0905049420519380402064586712958594039933889035337394627220130813347089
+718415056564330782186171899023705...
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