functions(2)

**Punch** · April 10th 2011, 03:00 AM

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.

**Prove It** · April 10th 2011, 03:15 AM

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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