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Math Help - Help with two log proof simultaneous questions please

  1. #1
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    [SOLVED] Help with two log proof simultaneous questions please

    Hi,

    My elderly father has been stumped by the following questions for the past few days. It's beyond my knowledge. I wondered if anyone here might be able to solve these two attached questions 10 and 16 (I did try to write them out and use LATEX code but to no avail)?

    Many thanks in advance.

    Here's my efforts using LATEX
    Q10: Show that log_{16} (xy)= \frac{1}{2} log_4 x+ \frac{1}{2} log_4 y. Hence, or otherwise, solve the simutaneous equations

    log_16 (xy)=3 \frac{1}{2}

    (( log_4 x) / ( log_4 y)) = -8

    Q16: Express log_9 x y in terms of log_3 x and log_3 y.
    Without using tables, solve for x and y the simultaneous equations

    log_9 x y= \frac{5}{2}
    log_3 x log_3 y = -6
    Attached Thumbnails Attached Thumbnails Help with two log proof simultaneous questions please-q10.jpg   Help with two log proof simultaneous questions please-q16.jpg  
    Last edited by Ahmadtaz; December 28th 2011 at 01:51 AM. Reason: Solved
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Help with two log proof simultaneous questions please

    To show that: \log_{16}(xy)=\frac{1}{2}\log_4(x)+\frac{1}{2}\log  _4(y)

    Rewrite:
    \frac{1}{2}\log_4(x)=\log_4(\sqrt{x})=\frac{\log_{  16}(\ldots)}{\log_{16}(\ldots)}

    Complete the \ldots and do exactly the same for the other term.

    Try to do something similar for the other question Q16.

    For the simultaneous equations: you have given two equations and two variables x and y so use the substitution method (do you know how this works?)
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  3. #3
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    Re: Help with two log proof simultaneous questions please

    Hi Siron,

    Many thanks for your speedy response. I've spoken to my Dad and he does understand what you are saying. He got as far as the following, however, its the Substitution method that he says doesn't seem to be giving him the result he expects.

    \frac{\log_4(x)}{\log_4(x)} = -8

    therefore \log_4(x) - \log_4(y) = -8


    \log_16(xy) = \frac{7}{2}
    therefore \frac{1}{2}\log_4(xy) = \frac{7}{2}
    therefore log_4(xy) = 7

    Can you please point out where he is going wrong? He knows that x = 4^(-1) = \frac{1}{4}, and he knows that y= 4^(8). He just can't work out how.

    On a personal note; I have downloaded the LATEXT how-to PDF but my previews have errors or look like the above. If you could tell me how you type LATEXT above correctly I would appreciate that.

    Thank you kindly for your help once again.
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  4. #4
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    Re: Help with two log proof simultaneous questions please

    Actually \frac{\log_4(x)}{\log_4(y)}=-8 gives \log_4(x)=-8\log_4(y) or x=y^{-8}.
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  5. #5
    MHF Contributor Siron's Avatar
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    Re: Help with two log proof simultaneous questions please

    To work with LaTEX you have to use the brackets: [ tex ] ... [ /tex ].
    We are given the simultaneous equations:
    \left \{ \begin{array}{ll} \log_{16}(xy)=\frac{7}{2} \\ \frac{\log_4(x)}{\log_4(y)}=-8 \end{array} \right .

    We can rewrite the first equation with the given:
    \left \{ \begin{array}{ll} \frac{1}{2}\left[\log_4(x)+\log_4(y)\right]=\frac{7}{2} \\ \frac{\log_4(x)}{\log_4(y)}=-8 \end{array} \right .

    Solving the system:
    \left \{ \begin{array}{ll} \frac{1}{2}\left[\log_4(x)+\log_4(y)\right]=\frac{7}{2} \\ \log_4(x)=-8\log_4(y) \end{array} \right .

    By substituting the second equation in the first we obtain:
    \left \{ \begin{array}{ll} \frac{1}{2}\left[-8\log_4(y)+\log_4(y)\right]=\frac{7}{2} \\ \log_4(x)=-8\log_4(y) \end{array} \right .

    \Leftrightarrow \ldots

    I guess you have switched the answers, beside my calculations I obtain: x=4^8 and y=\frac{1}{4} which is correct by substituting them into the system.

    Proceed.

    The other system of equations is similar to solve.
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  6. #6
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    Re: Help with two log proof simultaneous questions please

    Hi Siron and Plato,

    Many thanks for your help! My Dad seems to be very pleased with the information.

    Thank you kindly!
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  7. #7
    MHF Contributor Siron's Avatar
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    Re: Help with two log proof simultaneous questions please

    You're welcome!
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