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Math Help - Logarithm problem

  1. #1
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    Logarithm problem

    Q) Find the domain of the defintion of the function :-

    Y = log(1 - log(x^2 - 5x + 16))

    Any help would be greatly appreciated.

    Thanks,
    Ashish
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  2. #2
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    Quote Originally Posted by a69356 View Post
    Q) Find the domain of the defintion of the function :-

    Y = log(1 - log(x^2 - 5x + 16))

    Any help would be greatly appreciated.

    Thanks,
    Ashish
    Since the log-function is defined for positive real numbers only

    1 - \log(x^2 - 5x + 16)>0~\implies~\log(x^2 - 5x + 16)<1 ....and \log(x^2 - 5x + 16)>0

    De-logarithmize(?) both sides:

    x^2-5x+6<10~\implies~x^2+5x+6<0

    The LHS can be factored:

    (x+1)(x-6)<0 ......... Aproduct of 2 factors is negative (< 0) if one factor is negative and the other factor is positive:

    \begin{array}{rcl}x+1<0\wedge x-6>0 & ~\vee~ & x+1>0\wedge x-6<0 \\ x<-1\wedge x>6 &~\vee~& x>-1\wedge x<6 \\ \emptyset &~\vee~&\boxed{-1<x<6}\end{array}

    2nd condition:

    \log(x^2 - 5x + 16)>0~\implies~x^2-5x+16>1

    x^2-5x+15>0~\implies~x\in \mathbb{R}

    Therefore the domain of the function is d=(-1,6)
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