### 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 consider what values of x make the function undefined or invalid.
In general, the logarithm function $\log_b x$ is only defined for positive values of x and a positive base b that is not equal to 1.
So, for the innermost function $\log_4 x$, I need $x>0$ and $4>0$ and $4\neq 1$.
The last two conditions are trivially satisfied, so I only need to worry about $x>0$.
For the next function $\log_3(\log_4 x)$, I need $\log_4 x>0$ and $3>0$ and $3\neq 1$.
Again, the last two conditions are easy, so I only need to worry about $\log_4 x>0$.
But when is $\log_4 x>0$?
This is equivalent to saying $4^{\log_4 x}>4^0$, which simplifies to $x>1$ by the properties of logarithms.
So, for the second function to be defined, I need $x>1$.
Finally, for the outermost function $\log_2(\log_3(\log_4 x))$, I need $\log_3(\log_4 x)>0$ and $2>0$ and $2\neq 1$.
Once more, the last two conditions are obvious, so I only need to worry about $\log_3(\log_4 x)>0$.
But when is $\log_3(\log_4 x)>0$?
This is equivalent to saying $3^{\log_3(\log_4 x)}>3^0$, which simplifies to $\log_4 x>1$ by the properties of logarithms.
But when is $\log_4 x>1$?
This is equivalent to saying $4^{\log_4 x}>4^1$, which simplifies to $x>4$ by the properties of logarithms.
So, for the third function to be defined, I need $x>4$.
Putting it all together, I need $x>4$ for the entire function $f(x)$ to be defined.
This is the domain of the function.