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Math Help - Express and calculate...

  1. #1
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    Express and calculate...

    Hi,

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

    I got the expression part but not the calculation:

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

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

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

    \frac{x}{y} = \frac{3^{2a}}{3^b}
    = 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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    xy=243=3^5

    xy=3^{2a+b}

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

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

    \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
    xy=243=3^5

    xy=3^{2a+b}

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

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

    \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:

    2a + b = 5
    2a - b = 1

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

    2(\frac{3}{2}) - b = 1
    3 - b = 1
    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)

    3^1=3^{2a-b}

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

    3=3
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