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Math Help - weird log proof

  1. #1
    Member jacs's Avatar
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    weird log proof

    got his as a homework question and not sure if it is a misprint or this is the proper equation. Either way, got no clue how to proceed.

    2^x = 5^y = 10^z
    using logarithms to base 10 or otherwise, show that 1/z = 1/x + 1/y

    i am thinking one of those equal signs is really a plus maybe a minus???

    any help greatly appreciated.

    jacs
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by jacs View Post
    got his as a homework question and not sure if it is a misprint or this is the proper equation. Either way, got no clue how to proceed.

    2^x = 5^y = 10^z
    using logarithms to base 10 or otherwise, show that 1/z = 1/x + 1/y

    i am thinking one of those equal signs is really a plus maybe a minus???

    any help greatly appreciated.

    jacs
    Nope. The problem statement is correct. Let's use log base 10:

    2^x = 10^z

    log(2^x) = log(10^z) = z* log(10) = z

    x*log(2) = z

    or 1/x = log(2)/z

    Similarly: 1/y = log(5)/z

    Thus
    1/x + 1/y = log(2)/z + log(5)/z = [log(2) + log(5)]/z = log(2*5)/z = log(10)/z = 1/z (Check)

    -Dan
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    Quote Originally Posted by jacs View Post
    got his as a homework question and not sure if it is a misprint or this is the proper equation. Either way, got no clue how to proceed.

    2^x = 5^y = 10^z
    using logarithms to base 10 or otherwise, show that 1/z = 1/x + 1/y

    i am thinking one of those equal signs is really a plus maybe a minus???

    any help greatly appreciated.

    jacs
    Say that,
    2^x=t
    5^y=t
    10^z=t
    Then,
    x log 2=log t
    y log 5=log t
    z=log t
    Thus,
    1/z=1/log t
    1/x=log 2/log t
    1/y=log 5/log t

    But,
    1/x+1/y=log 5/log t+log 2/log t=(log 2+log 5)/log t
    Note the numerator is the identity for logarithm product,
    log (2*5)/log t=log 10/log t=1/log t=1/z

    This is mine 26th post!!!
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