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 Fortieth logs (Posted on 2011-05-31)
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

 Subject Author Date re(3): Who is right for 7?....DETAILS Ady TZIDON 2011-06-02 11:43:53 re(2): Who is right for 7?....DETAILS Gamer 2011-06-02 05:11:42 re(2): Closer number, closing remarks Ady TZIDON 2011-06-02 01:59:01 re: Closer number Steve Herman 2011-06-02 00:42:38 re(2): Who is right for 7? Ady TZIDON 2011-06-01 20:39:15 re: Who is right for 7?....DETAILS Ady TZIDON 2011-06-01 19:23:47 Answers from 2 to 9 Math Man 2011-06-01 19:05:31 Closer number Math Man 2011-06-01 19:01:51 re: Who is right for 7? JayDeeKay 2011-06-01 18:41:41 Who is right for 7? Jer 2011-06-01 15:08:08 Lucky 7 Steve Herman 2011-05-31 21:13:59 No Subject JayDeeKay 2011-05-31 17:57:15 re(2): A start (spoiler) Jer 2011-05-31 17:06:27 re: A start (spoiler) Charlie 2011-05-31 14:43:51 A start (spoiler) Steve Herman 2011-05-31 14:12:29

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