Determine the domain: f(x) = \frac{ln(ln(x + 1))}{e {}^(x) - 9 }

Question

asked Nov 19, 2023 206k views

1 Answer

Jannes Carpentier · Answer 1 · 2023-11-25T11:20:57+0000

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

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

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