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How to compare powers

  1. Dec 31, 2015 #1
    This is by no means a trick question, I would want to know whether there is any technique to compare two powered terms with different bases, that is if I say I want to compare 2 powered 200 to 3 powered 120 to estimate which of these may be greater without any special or digital aid, how can I do it in my mind?
     
  2. jcsd
  3. Dec 31, 2015 #2

    TeethWhitener

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    Gold Member

    Take the logarithm of both numbers:

    ln(2200) = 200 ln 2 = 138.629436112
    ln(3120) = 120 ln 3 = 131.83347464

    So 2200 > 3120

    EDIT: fixed a math error.
     
  4. Dec 31, 2015 #3
    Are you talking about exponents?
    So like x^n vs y^m?
    You could just take the logs, and do some algebra.

    Have the ~ symbol be a placeholder for =,<,> symbols

    [tex] 2^{200} \sim 3^{120} [/tex]
    [tex] 200 log(2) \sim 120 log(3) [/tex]
    [tex] \frac{200}{120} \sim \frac{log(3)}{log(2)} [/tex]
    [tex] \frac{5}{3} \sim log_2(3) [/tex]
    [tex] 2^{\frac{5}{3}} \sim 3 [/tex]
    [tex] 2^5 \sim 3^3 [/tex]
    [tex] 32 \sim 27 [/tex]

    therefore: ~ is >
    so
    [tex] 2^{200} > 3^{120} [/tex]

    No calculator needed.
     
  5. Dec 31, 2015 #4
    Thanks.
     
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