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Math Help - Logarithm inequalities

  1. #1
    xwy
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    Logarithm inequalities

    Hello, I've trouble solving "graph log2x<0" how do i go about doing this? I know that log2x=y is equivalent to 2^y=x. However, the inequality sign and the <0 makes it complicated. The given answer is 1>x>0.
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    Re: Logarithm inequalities

    \log_2(x) = \dfrac{\ln x}{\ln 2}

    If y>0, then 2^{-y} = \dfrac{1}{2^y}. If you increase y, \dfrac{1}{2^y} gets "closer" to zero. You should know that there is no value for y that makes the equation 2^y = 0 true, so \log_2 (0) is not defined. Thus, x>0. Instead of increasing y, let's decrease it. As y gets closer to zero, 2^{-y} gets closer to 1, but is always less than one. So, that tells you x<1.
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    Re: Logarithm inequalities

    You graph this inequality the exact same way you graph ANY inequality.

    First, you graph the related equality. In this case that seems to be $y = log_2(x).$ Is that function defined for non-positive values of x?

    Second, you identify on what side of the point or points of equality, the inequality is true.

    So at what value of x does $log_2(x) = 0.$

    To the left of that point what is the sign of the function? To the right of that point what is the sign of the function?
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    Re: Logarithm inequalities

    Quote Originally Posted by xwy View Post
    Hello, I've trouble solving "graph log2x<0" how do i go about doing this? I know that log2x=y is equivalent to 2^y=x. However, the inequality sign and the <0 makes it complicated. The given answer is 1>x>0.
    Well the domain of $\displaystyle \begin{align*} \log_2{(x)} \end{align*}$ is $\displaystyle \begin{align*} x > 0 \end{align*}$. Solving the inequality for x gives

    $\displaystyle \begin{align*} \log_2{(x)} &< 0 \\ x &< 2^0 \\ x &< 1 \end{align*}$

    Thus, when combined with the domain of this function, the solution must be $\displaystyle \begin{align*} 0 < x < 1 \end{align*}$.
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    Re: Logarithm inequalities

    This has only the single variable x, rather than, say x and y. Its graph will be a portion of a number line, not a two dimensional x, y graph. In general, the regions where f(x)< a and f(x)> a are separated by points where f(x)= a. Here, mark the single point x such that log_2(x)= 1. Then, perhaps, choose one value of x less than that and one value of x larger to determine which side satisfies log_2(x)< 1 and which satisfies k=log_2(x)> 1.
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