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Math Help - Determing range of values of x & y

  1. #1
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    Determing range of values of x & y

    Hi

    If real variables x and y are related by the equation:

    ln(3-y) = 3ln(2x+1)-x

    then how do i:

    a). determine the range of values of x and y for which expressions on each side of equation are defined, and

    b). find y explicitly as a function of x, that is, express the equation in the form y=f(x)

    Thanks
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  2. #2
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    Quote Originally Posted by moolimanj View Post
    Hi

    If real variables x and y are related by the equation:

    ln(3-y) = 3ln(2x+1)-x

    then how do i:

    a). determine the range of values of x and y for which expressions on each side of equation are defined, and

    b). find y explicitly as a function of x, that is, express the equation in the form y=f(x)

    Thanks
    to a) The ln-function is defined for positive arguments:

    3-y>0~\implies~y<3 and

    2x+1>0~\implies~x>-\frac12

    To b)

    \ln(3-y) = 3\ln(2x+1)-x~\implies~3-y=\frac{(2x+1)^3}{e^x}~\implies~y=3- \frac{(2x+1)^3}{e^x}~,~x>-\frac12
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  3. #3
    MHF Contributor red_dog's Avatar
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    a) The conditions are
    3-y>0
    2x+1>0
    Now, can you solve them?

    b) The equation can be written as
    \displaystyle\ln(3-y)=\ln(2x+1)^3-\ln e^x\Leftrightarrow\ln(3-y)=\ln\frac{(2x+1)^3}{e^x}\Leftrightarrow
    \displaystyle\Leftrightarrow 3-y=\frac{(2x+1)^3}{e^x}\Leftrightarrow y=3-\frac{(2x+1)^3}{e^x}
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