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Proof on logarithms

  1. Jun 5, 2016 #1
    1. The problem statement, all variables and given/known data
    Prove that ##{1/3} < log_{34} 5 < {1/ 2}##

    2. Relevant equations
    ##log_b a = {1/ log_a b}##
    ##logmn = logm + logn##


    3. The attempt at a solution
    ##log_{34} 5 = {1/ log_5 34}##
    ##= 1/(log_5 17 + log_5 2)##
    ##=1/(1 + log_5 3 + log_5 2 + something)##
    ##=1/(1 + log_5 6 + something)##
    ##=1/(2+something~else)##

    something else ##<1##

    Hence it is greater than 1/3 and smaller than 1/2.
    But I think mathematically speaking, that ##something## isn't supposed to be there.
     
  2. jcsd
  3. Jun 5, 2016 #2
    A better way to do this is to recast the inequality as
    [tex]3 > \log_{5}34 > 2[/tex]
    and rewrite ##3## and ##2## in terms of base-5 logarithms as well.
     
  4. Jun 5, 2016 #3
    Thanks. Made it much easier than my solution.
     
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