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Math Help - functions(2)

  1. #1
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    functions(2)

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

    \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 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 e^x - 2 > 1.

    Solve for \displaystyle x.
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