Finding the domain of $\log(\log(x))$ Why is the domain of $f(x)=\log(x)$ is $x&gt;0$ but the domain of $f(x)=\log\log(x)$ is $x&gt;1$. 

Why is there a difference? (default value of base is 10)
If $f(x)=\log(\log x)$, then it has to be $x&gt;0$, but the argument of the most external logarithm, argument which is $\log x$ also has to be greater than $0$.

So $\log x&gt;0$ is equivalent to $x&gt;10^0$, that is $x&gt;1$. Then you need both $x&gt;0$ and $x&gt;1$ to be true, and you can just say that $x&gt;1$. Then, $(1,+\infty)$ is the domain of $f$.
The domain of the function $g(x)=\log_{10}(f(x))$ is the set of all $x$ such that $f(x)&gt;0$. In particular, if $f=\log_{10}$ then $$\log_{10}(x)&gt;0 \quad \textrm{iff} \quad x&gt;10^0 =1$$