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Math Help - How would I simplify this?

  1. #1
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    How would I simplify this?

    Log\sqrt{x^3-3x}=\frac{1}{2}

    I got to: 10^\frac{1}{2}=(x^3-3x)^\frac{1}{2} and then got a little confused on how I would distribute the exponent and solve it.

    Any and all pointers highly appreciated, thanks!
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  2. #2
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    e^(i*pi)'s Avatar
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    Using Log laws:

    \log(\sqrt{x^3-3x}) = \dfrac{1}{2}\log(x^3-3x)

    \log(x^3-3x) = 1

    x^3-3x = 10

    Can you continue? Bear in mind x > 0

    ===================

    Using your method you can square both sides to give 10 = x^3-3x which goes to show there's more than one way to skin a cat. Since we know that x>0 we needn't worry about extraneous solutions
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  3. #3
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    Quote Originally Posted by e^(i*pi) View Post
    Using Log laws:

    \log(\sqrt{x^3-3x}) = \dfrac{1}{2}\log(x^3-3x)

    \log(x^3-3x) = 1
    Hmm, do you mind showing me how you got to \log(x^3-3x) = 1?

    Quote Originally Posted by e^(i*pi) View Post
    Using your method you can square both sides to give 10 = x^3-3x which goes to show there's more than one way to skin a cat. Since we know that x>0 we needn't worry about extraneous solutions
    Ah, that would work. Thanks a lot!
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  4. #4
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by youngb11 View Post
    Hmm, do you mind showing me how you got to \log(x^3-3x) = 1?



    Ah, that would work. Thanks a lot!
    Certainly

    You have the equation \log(\sqrt{x^3-3x}) = \dfrac{1}{2}. By the laws of logarithms [\ln(a^k) = k\ln(a)] so it can be as \dfrac{1}{2}\log(x^3-3x) = \dfrac{1}{2}.

    Either by inspection or multiplying by 2 will cancel the 1/2 on both sides to give \log(x^3-3x) = 1
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  5. #5
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    Quote Originally Posted by e^(i*pi) View Post
    Certainly

    You have the equation \log(\sqrt{x^3-3x}) = \dfrac{1}{2}. By the laws of logarithms [\ln(a^k) = k\ln(a)] so it can be as \dfrac{1}{2}\log(x^3-3x) = \dfrac{1}{2}.

    Either by inspection or multiplying by 2 will cancel the 1/2 on both sides to give \log(x^3-3x) = 1
    Thanks a lot for the clear explanation! Have a great day
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