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2020-01-24, 13:47   #15
Dr Sardonicus

Feb 2017
Nowhere

63028 Posts

Quote:
 Originally Posted by Happy5214 "floor" (for positive numbers) just means to truncate the part after the decimal point. It really should be the ceiling (think of 100, which has 3 digits and a base-10 log of 2). The multiplication by (log(2) / log(10)) converts the exponent (which is the base-2 log of the 2^n part) to the base-10 log. The first 2 (I think) was supposed to be the k value (10), which really should have been 1. Replace that with the base-10 log of whatever k is (e.g. log(231) / log(10) for your last example). The -1 at the end can basically be ignored.
Actually, I meant 2. Generally, the number of decimal digits in N is 1 + floor(log(N)/log(10)). If N = 10*2^13509 - 1, then (since N+1 is not a power of ten), N has the same number of decimal digits as N+1, so we may use 1 + floor(log(N+1)/log(10)). Now

log(N+1) = log(10) + 13509*log(2), so we have

1 + floor(log(N+1)/log(10)) = 2 + floor(13509*log(2)/log(10)), which is 4068.

Siccing the mighty Pari-GP on the previous query,

Code:
? 1+floor((log(231) + 233528*log(2))/log(10))

%2 = 70302
As a ballpark estimate, log(2)/log(10) is close to 0.3 (more closely, 0.30102999).

Also, I meant floor(), the integer floor. The integer ceiling of an exact integer is that integer, so it gives the wrong answer for exact powers of ten; ceil(log(100)/log(10)) is 2, not 3.