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Thread: Express and calculate...

  1. #1
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    Express and calculate...

    Hi,

    If $\displaystyle \log_9 x = a$ and $\displaystyle \log_3 y = b$, express $\displaystyle xy$ and
    $\displaystyle \frac{x}{y}$ as powers of $\displaystyle 3$.
    If $\displaystyle xy = 243$
    and $\displaystyle \frac{x}{y} = 3$, calculate $\displaystyle a$ and $\displaystyle b$.

    I got the expression part but not the calculation:

    $\displaystyle \log_9 x = a,\,\,\,\,\,\,\,\,\,\,\log_3 y = b$

    $\displaystyle x = 9^a = 3^{2a}\,\,\,\,\,\,\,\,y = 3^b$

    $\displaystyle xy = 3^{2a} \times 3^b$
    $\displaystyle xy = 3^{2a + b}$

    $\displaystyle \frac{x}{y} = \frac{3^{2a}}{3^b}$
    $\displaystyle = 3^{2a - b}$
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  2. #2
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    What don't you get? It looks fine.
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  3. #3
    MHF Contributor harish21's Avatar
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    $\displaystyle xy=243=3^5$

    $\displaystyle xy=3^{2a+b}$

    $\displaystyle \therefore 3^5=3^{2a+b}$...............(I)

    likewise, $\displaystyle \frac{x}{y}=3 \;and\;\frac{x}{y}=3^{2a-b}$

    $\displaystyle \therefore 3^{1}=3^{2a-b}$................(II)

    you can find a and b from equations (I) and (II)
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  4. #4
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    Quote Originally Posted by harish21 View Post
    $\displaystyle xy=243=3^5$

    $\displaystyle xy=3^{2a+b}$

    $\displaystyle \therefore 3^5=3^{2a+b}$...............(I)

    likewise, $\displaystyle \frac{x}{y}=3 \;and\;\frac{x}{y}=3^{2a-b}$

    $\displaystyle \therefore 3^{1}=3^{2a-b}$................(II)

    you can find a and b from equations (I) and (II)
    Ah, I made an attempt like that but I had no faith in it. Thanks.

    I'm not sure if this works:

    $\displaystyle 2a + b = 5$
    $\displaystyle 2a - b = 1$

    $\displaystyle 4a = 6$
    $\displaystyle a = \frac{3}{2}$

    $\displaystyle 2(\frac{3}{2}) - b = 1$
    $\displaystyle 3 - b = 1$
    $\displaystyle b = 2$
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  5. #5
    MHF Contributor harish21's Avatar
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    -Well Done!

    You can check whether your values for a and b are correct or not by plugging their values back in equations (I) and (II)

    Check:

    Equation (II)

    $\displaystyle 3^1=3^{2a-b}$

    $\displaystyle 3=3^{2\cdot\frac{3}{2}-2} $

    $\displaystyle 3=3$
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