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Math Help - How to find differential of function of two variables : LOG[y,x] ?

  1. #1
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    Lightbulb How to find differential of function of two variables : LOG[y,x] ?

    I have function: z=(Log[y,x])^2 where y is base of log.
    I need to find differential of function z regarding the x.
    My solution:
    [f(g(x))]'=f'(g(x))*g'(x)
    so...
    [(Log[y,x])^2]'=(2*Log[y,x])*(1/(x*Ln[y]))
    but math program gives me following answer:
    2Ln[x]/(x*Ln[y]^2)
    What is the correct answer? Is there any mistake?
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  2. #2
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    Quote Originally Posted by RCola View Post
    I have function: z=(Log[y,x])^2 where y is base of log.
    I need to find differential of function z regarding the x.
    My solution:
    [f(g(x))]'=f'(g(x))*g'(x)
    so...
    [(Log[y,x])^2]'=(2*Log[y,x])*(1/(x*Ln[y]))
    but math program gives me following answer:
    2Ln[x]/(x*Ln[y]^2)
    What is the correct answer? Is there any mistake?
    Hi RCola.

    If I understand correctly, you have z=\left(\log_{\,y}x\right)^2, and you need to find \displaystyle {{\partial z}\over{\partial x}}\,.

    The change of base formula gives: \displaystyle \log_{\,B} A = {{\ln A}\over{\ln B}}\ .

    \displaystyle z=\left({{\ln x}\over{\ln y}}\right)^2

    BTW: I believe the answers are equivalent AND both are correct.
    Last edited by SammyS; February 17th 2011 at 07:02 AM.
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  3. #3
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    Use the change of base rule to get your log into base e:

    z = (\log_y(x))^2 = \left(\dfrac{\ln(x)}{\ln(y)}\right)^2

    Treat (\ln(y))^2 as constant (because we're differentiating with respect to x)

    \dfrac{dz}{dx} = \dfrac{\frac{d}{dx} [\ln(x)]^2}{(\ln(y))^2}

    Use the chain rule to differentiate the numerator

    Since this gives the answer the book gives I don't think you have to implicitly differentiate
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