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Math Help - log help

  1. #1
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    log help

    Been trying to figure this out but haven't had any luck

    There is eactly one real number k for which the equation
    (log3x)(log5x) =k
    has only one real solution for x. What is the value of x.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by floridaboatkid
    Been trying to figure this out but haven't had any luck

    There is eactly one real number k for which the equation
    (log3x)(log5x) =k
    has only one real solution for x. What is the value of x.
    I am guessing but I would bet ready money that your question should
    read somethink like:

    "There is eactly one real number k for which the equation
    (log_3(x))(log_5(x)) =k
    has only one real solution for x. What is the value of x"

    Where log_b denotes the log to base b.

    So lets assume your question is:

    "There is eactly one real number k for which the equation
    log_3 (x).log_5 (x)\ =\ k
    has only one real solution for x. What is the value of x"

    as log_b(x)\ =\ log_e(x)/log_e(b) your equation becomes:

    \frac{log_e (x).log_e (x)}{log_e(3).log_e(5)}\ =\ k

    or:

     (log_e(x))^2\ =\ k.log_e(3).log_e(5)

    Then the condition of the problem implies that k\ =\ 0
    as otherwise there will be two solution for x, so:

    log_e(x)\ =\ 0

    or
    x\ =\ 1

    Quite Easily Done (aka QED )

    RonL
    Last edited by CaptainBlack; November 21st 2005 at 01:51 AM.
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  3. #3
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    no they are base 10

    the equation is written in log base 10
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by floridaboatkid
    the equation is written in log base 10
    OK, then lets rewrite your problem so that it is unambiguous:

    There is exactly one real number k for which the equation:
    (log_{10}(3x))(log_{10}(5x)) =k
    has only one real solution for x. What is the value of x.
    Now this equation only makes sense with:
    x\ \epsilon\ (0,\ \infty).
    Also as:
    x\ \rightarrow\ 0\ ,\ (log_{10}(3x))(log_{10}(5x))\ \rightarrow\ \infty,
    and
    x\ \rightarrow\ \infty\ ,\ (log_{10}(3x))(log_{10}(5x))\ \rightarrow\ \infty.

    So an equation:
    (log_{10}(3x))(log_{10}(5x))\ =\ k
    has either multiple solutions, zero solutions, or when k is the global
    minimum of (log_{10}(3x))(log_{10}(5x)) exactly one solution.

    So your problem becomes find the x which corresponds to the global minimum of:

    y(x)\ =\ (log_{10}(3x))(log_{10}(5x))

    RonL
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  5. #5
    Grand Panjandrum
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    Now our problem is to find the global minimum of:

    <br />
y(x)\ =\ (log_{10}(3x))(log_{10}(5x))<br />

    The first thing we do is to sketch the graph of this
    function. A couple of rather pathetic Excel plots of
    this are shown in the attachment to this post.

    From these plots it is clear that the minimum we seek
    is the sort which is obtained by differentiating and
    setting the resulting derivative to zero, that is we seek
    the solution of:

    <br />
\frac{dy}{dx}\ =\ 0<br />

    which should be a routine task, which I will leave to you.

    RonL
    Attached Thumbnails Attached Thumbnails log help-plot.gif  
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