### Instruction: What is the domain of the function $f(x)=\log_2(\log_3(\log_4x))$?### Response: To find the domain of a function, I need to figure out what values of $x$ make the function well-defined and real-valued.
I know that the logarithm function $\log_b(a)$ is only defined for positive values of $a$ and $b$, and it is only real-valued when $b$ is not equal to 1.
So, for the innermost function $\log_4x$, I need to have $x>0$ and $4\neq 1$.
The second condition is always true, so I only need to worry about the first one.
For the next function $\log_3(\log_4x)$, I need to have $\log_4x>0$ and $3\neq 1$.
The second condition is again always true, so I only need to worry about the first one.
To have $\log_4x>0$, I need to have $x>1$, since $\log_4(1)=0$ and the logarithm function is increasing.
For the outermost function $\log_2(\log_3(\log_4x))$, I need to have $\log_3(\log_4x)>0$ and $2\neq 1$.
The second condition is yet again always true, so I only need to worry about the first one.
To have $\log_3(\log_4x)>0$, I need to have $\log_4x>1$, since $\log_3(1)=0$ and the logarithm function is increasing.
To have $\log_4x>1$, I need to have $x>4$, since $\log_4(4)=1$ and the logarithm function is increasing.
So, putting it all together, I need to have $x>4$ for the function $f(x)$ to be well-defined and real-valued.
This is the domain of the function.