How would you fin the integer closest to log(log(1234567891011121314...2013)) where the number

taghdh9

taghdh9

Answered question

2022-06-28

How would you fin the integer closest to
log(log(1234567891011121314...2013))
where the number is the concatenation of numbers 1 through 2013 inclusive. log() in this case is log base 10.
Also, how would you find the remainder when it is divided by 75?

Answer & Explanation

assumintdz

assumintdz

Beginner2022-06-29Added 22 answers

In general, if a positive integer n and d digits in its decimal expansion then
d 1 log ( n ) d
For example log ( 1529 ) = 3.1844 and 1529 has 4 digits.
How many digits are in 12345678 2013 ? Well we have
9 one-digit numbers
99 9 = 90 two-digit numbers
999 99 = 900 three-digit numbers
2013 999 = 1014 four-digit numbers
So your number has 9 × 1 + 90 × 2 + 900 × 3 + 1014 × 4 = 6945 digits.
So we must have
6944 log ( 12345678 2013 ) 6945
and so log ( 6944 ) log ( log ( 12345678 2013 ) ) log ( 6945 )
Can you take it from here?

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