Finding the domain of a natural log composite function

Question

If $f(x) = ln(x)$, what is the largest possible domain of $f(f(x))$?

I only know that the input for $ln(x)$ must always be greater than 0, so the domain must be $x$ > $0$ for $ln(x)$ alone. However, once I plug in $f(x)$ into $f(x)$ to get $f(f(x)) = ln(ln(x))$, I'm not sure how to approach this problem as I believe I have to solve $ln(ln(x)) > 0$.

Answer 1

You have correctly identified that the domain of $\ln(x)$ is $x>0$.

As $\ln(x)$ is only defined for positive $x$ values, the domain of the composition

$$\ln(\ln (x))$$

will be where $\ln (x) > 0$. We have

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

Therefore, the domain of $\ln(\ln (x))$ is $x>1$.

Answer 2

Let $y=\ln x$ then $\ln(\ln x)=\ln y$.

$\ln y$ makes sense only if $y>0$, i.e. $\ln x>0$, i.e $x>1$.

So the largest domain for $\ln(\ln x)$ is $x>1$.

Answer 3

Generally, for any composition of two functions $h(x) = f(g(x))$, the domain of $h$ is the solution set of $g(x)\in D$, where $D$ is the domain of $f$. In your example, $D$ is the domain of $ln(x)$, which means $x>0$. Then $g(x)\in D$ means $x> 1$.