2
^{10} = 1024
2
^{20} = 1048576
Note that raising 2 to each of the first two multiples of 10 results in a number whose first digit is 1.
Find the smallest multiple of 10 where 2 raising to that power results in a number that does not begin with 1.
The mantissa of log(1024) is .010299956639812 (common logs), while that of 2 is .3010299956639811; dividing the first into the second gives 29.22633620615765, so that 1024^29 will still begin with 1 but 1024^30 will begin with 2.
Checking via calculator app:
1024^29 ~= 1.989292945639148 * 10^87
1024^30 ~= 2.037035976334467 * 10^90
via UBASIC, these two numbers are:
1989292945639146568621528992587283360401824603189390869761855907572637988050133502132224
2037035976334486086268445688409378161051468393665936250636140449354381299763336706183397376
The powers of 2 of course are 10 times the powers of 1024: 290 and 300, the latter being the sought answer.

Posted by Charlie
on 20150415 09:01:19 