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Thread: functions(2)

  1. #1
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    functions(2)

    given the functions $\displaystyle g(x)=e^x-2, x=(a,infinity)$
    $\displaystyle
    h(x)=ln(lnx), x>1
    $
    Find the smallest value of $\displaystyle a$ in exact form such that the composite function $\displaystyle hg$ exists.
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  2. #2
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    Assuming that you are trying to find $\displaystyle \displaystyle h\left(g(x)\right)$...

    $\displaystyle \displaystyle h\left(g(x)\right) = \ln{\left[\ln{\left(e^x - 2\right)}\right]}$.

    A logarithm is only defined for positive values of the independent variable, so the innermost logarithm can only be evaluated where $\displaystyle \displaystyle e^x - 2 > 0$.

    But since another logarithm will be taken, you can only accept those values for which the innermost logarithm is positive. So where $\displaystyle \displaystyle e^x - 2 > 1$.

    Solve for $\displaystyle \displaystyle x$.
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