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Fortieth logs (Posted on 2011-05-31) Difficulty: 2 of 5
It turns out that the common logarithms of each of the numbers from 2 through 9 can be very well approximated by rational numbers of the form n/40.

Derive each of the numerators with no calculation aids beyond pencil and paper.

  Submitted by Jer    
Rating: 4.5000 (2 votes)
Solution: (Hide)
To begin note that 210 = 1024 which is very close to 1000 = 103.
210 ≈ 103
10 log 2 ≈ 3
log 2 ≈ 3/10 = 12/40

For 3, note that 34 = 81 which is very close to 80.
34 ≈ 80 = 10*23
4 log 3 ≈ 1 + 3 log2 ≈ 76/40
log 3 ≈ 19/40

Since 4 = 22, log 4 = 2 log 2 ≈ 2*12/40 = 24/40

Since 5 = 10/2, log 5 = log 10 - log 2 ≈ 1 - 12/40 = 28/40

Since 6 = 2*3, log 6 = log 2 + log 3 ≈ 12/40 + 19/40 = 31/40

For 7, note 72 = 49 which is close to 50 = 100/2
log 72 = log 100/2
2 log 7 = 2 - log 2 ≈ 2 - 12/40 = 68/40
log 7 ≈ 34/40,
see the comments for an argument about why 33/40 is also pretty good.

Since 8 = 23, log 8 = 3 log 2 ≈ 3*12/40=36/40

Since 9 = 33, log 9 = 2 log 3 ≈ 2*19/40=38/40

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Some Thoughtsre(3): Who is right for 7?....DETAILSAdy TZIDON2011-06-02 11:43:53
re(2): Who is right for 7?....DETAILSGamer2011-06-02 05:11:42
Some Thoughtsre(2): Closer number, closing remarksAdy TZIDON2011-06-02 01:59:01
Some Thoughtsre: Closer numberSteve Herman2011-06-02 00:42:38
re(2): Who is right for 7?Ady TZIDON2011-06-01 20:39:15
Hints/Tipsre: Who is right for 7?....DETAILSAdy TZIDON2011-06-01 19:23:47
SolutionAnswers from 2 to 9Math Man2011-06-01 19:05:31
Closer numberMath Man2011-06-01 19:01:51
re: Who is right for 7?JayDeeKay2011-06-01 18:41:41
Who is right for 7?Jer2011-06-01 15:08:08
Lucky 7Steve Herman2011-05-31 21:13:59
No SubjectJayDeeKay2011-05-31 17:57:15
re(2): A start (spoiler)Jer2011-05-31 17:06:27
re: A start (spoiler)Charlie2011-05-31 14:43:51
Some ThoughtsA start (spoiler)Steve Herman2011-05-31 14:12:29
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