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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- Apr 10th 2011, 03:00 AMPunchfunctions(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. - Apr 10th 2011, 03:15 AMProve It
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 $\displaystyle \displaystyle x$.