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Thread: laws of logarithms

  1. Oct 28th 2006, 07:19 AM #1
    Yogi_Bear_79
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    laws of logarithms

    Please help with this example to apply the laws of logarithms to simplify the function f(x) =ln \sqrt\frac{9-x^2}{4+x^2}. Then find it’s derivative.
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  2. Oct 28th 2006, 07:27 AM #2
    Quick
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    Quote Originally Posted by Yogi_Bear_79 View Post
    Please help with this example to apply the laws of logarithms to simplify the function f(x) =ln \sqrt\frac{9-x^2}{4+x^2}. Then find it’s derivative.
    I can't help you with the derivative, but here's the simply part:

    you have: \ln\sqrt{\frac{9-x^2}{4+x^2}}

    rewrite: \ln\left(\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}}\right)

    Thus: \frac{1}{2}\ln\frac{9-x^2}{4+x^2}

    rewrite: \frac{1}{2}\ln\frac{(3+x)(3-x)}{4+x^2}

    It is possible to go farther but I don't know how that might help...
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  3. Oct 28th 2006, 07:35 AM #3
    Soroban
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    Hello, Yogi_Bear_79!

    Do you know the Laws of Logarithms?

    Apply the laws of logarithms to simplify the function f(x) = \ln\sqrt\frac{9-x^2}{4+x^2}
    Then find its derivative.

    We have: . f(x) \;= \;\ln\sqrt{\frac{9-x^2}{4+x^2}} \:=\:\ln\!\left(\frac{9-x^2}{4+x^2}\right)^{\frac{1}{2}} \;=\;\frac{1}{2}\cdot\ln\!\left(\frac{9-x^2}{4+x^2}\right)

    Then: . f(x)\;=\;\frac{1}{2}\bigg[\ln(9-x^2) - \ln(4 + x^2)\bigg]


    Now differentiate it . . .
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  4. Oct 28th 2006, 07:37 AM #4
    Quick
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    Quote Originally Posted by Soroban View Post




    Then: . f(x)\;=\;\frac{1}{2}\bigg[\ln(9-x^2) - \ln(4 + x^2)\bigg]

    Wait, so \log\frac{a}{b}=\log a-\log b? That's useful...
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  5. Oct 28th 2006, 11:23 AM #5
    topsquark
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    Quote Originally Posted by Quick View Post
    Wait, so \log\frac{a}{b}=\log a-\log b? That's useful...
    Yes it's true and yes it's useful!

    -Dan
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  6. Oct 28th 2006, 12:04 PM #6
    earboth
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    Quote Originally Posted by Quick View Post
    ...That's useful...
    Hi,

    I've attached a pdf-file with the logarithm rules and the corresponding power rules. Maybe this is useful for you too.

    EB
    Attached Files Attached Files
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