Question

Determine the domain:20%7B%7D%5E%7Bx%7D%20-%209%20%7D%20" id="TexFormula1" title="f(x) = \frac{ln(ln(x + 1))}{e {}^{x} - 9 } " alt="f(x) = \frac{ln(ln(x + 1))}{e {}^{x} - 9 } " align="absmiddle" class="latex-formula">
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Answer 1

The denominator cannot be zero, so

$e^x - 9 = 0 \implies e^x = 9 \implies x = \ln(9)$

is not in the domain of $f(x)$.

$\ln(x)$ is defined only for $x>0$, and we have

$\ln(x+1) > 0 \implies e^{\ln(x+1)} > e^0 \implies x+1 > 1 \implies x>0$

so there is no issue here.

By the same token, we need to have

$x+1 > 0 \implies x > -1$

Taking all the exclusions together, we find the domain of $f(x)$ is the set

$\left\{ x \in \Bbb R \mid x > 0 \text{ and } x \neq \ln(9)\right\}$

or equivalently, the interval $(0,\ln(9))\cup(\ln(9),\infty)$.