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Math Help - Lovely Logarithms

  1. #1
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    Smile Lovely Logarithms

    Hello,
    I tried to solve this question, but I got the conclusion that I couldn't.
    Therefore I would appreciate your help

    The problem is:

    Let log10P = x, log10Q = y and log10R = z.
    Express log10 (P/QR^3)^2 in terms of x, y and z.

    Thanks.
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  2. #2
    MHF Contributor red_dog's Avatar
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    Use this:

    \lg (a\cdot b)=\lg a+\lg b
    \lg\left(\frac{a}{b}\right)=\lg a-\lg b
    \lg a^n=n\lg a
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  3. #3
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    Thanks for your help,
    but how?
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  4. #4
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    Hello, Aminekhadir!


    "How?" . . . Have you never worked a log problem? . . . ever?

    Okay, just this once . . . I'll baby-step through it for you.


    Since all the logs are base-ten, I'll drop the base . . .


    Let: . \log(P) = x,\;\;\log(Q)= y,\;\;\log(R)= z

    Express  \log\bigg(\frac{P}{QR^3}\bigg)^2 in terms of x, y\text{ and }z.

    We have: . \log\bigg(\frac{P}{QR^3}\bigg)^2 \;=\;2\bigg[\log\left(\frac{P}{QR^3}\right)\bigg]

    . . =\;2\bigg[\log(P) - \log(QR^3)\bigg]

    . . = \;2\bigg[\log(P) - \left\{\log(Q) + \log(R^3) \right\} \bigg]

    . . =\;2\bigg[\log(P) - \log(Q) - \log(R^3)\bigg]

    . . = \;2\bigg[\underbrace{\log(P)}_x - \underbrace{\log(Q)}_y - 3\underbrace{\log(R)}_z\bigg]

    . . = \;2\left(x - y - 3z\right) \quad\hdots\quad\text{or: }\;2x - 2y - 6z

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