text: 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
  
  Beginner2022-06-29Added 22 answers
  
  In general, if a positive integer $n$ and $d$ digits in its decimal expansion then
  $d-1\le \mathrm{log}\left(n\right)\le d$
  For example $\mathrm{log}\left(1529\right)=3.1844\dots$ and $1529$ has $4$ digits.
  How many digits are in $12345678\dots 2013$ ? 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
  $6944\le \mathrm{log}\left(12345678\dots 2013\right)\le 6945$
  and so $\mathrm{log}\left(6944\right)\le \mathrm{log}\left(\mathrm{log}\left(12345678\dots 2013\right)\right)\le \mathrm{log}\left(6945\right)$
  Can you take it from here?
  
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