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Base 10 log question

  1. Sep 29, 2008 #1
    Is there a way to calculate multi digit base 10 logs without a calc or tables?

    eg.. what is ? log (base 10) of 1381.42?? (without a calulator)??
     
  2. jcsd
  3. Sep 29, 2008 #2
    (with caveat that i memorized log 1 - 9)
     
  4. Sep 30, 2008 #3

    lurflurf

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    Homework Helper

    Of coursethat is possible, the first people who made tables did not have calculators.
    since
    log(ab)=log(a)+log(b)
    numbers should be decomposed multiplicatively.
    One method is to factor into primes
    log(1381.42)=-2+log(2)+2log(17)+log(239)
    This shows a problem logs of many primes will be needed
    another method would involve writing the number as a product of factors with known logs and factors near 1 since when x is small
    log(1+x)~log(e)[x-x^2/2+x^3/3-x^4/4+...]
    say one knew log(138176)
    then
    log(1381.42)=-2+log(138176)-log(1+1/4064)
     
  5. Sep 30, 2008 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    What I would do is this:
    First, log(1381.42)= log(0.138142 x104)= 4+ log(0.138142).

    Now, use the 'Taylor's Polynomial' expansion:
    for 0< x< 1, log(x)= x+ (1/2)x2+ (1/3)x^3+ ...+ (1/n)x^n.
     
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