### Instruction: What is the domain of the function $f(x)=\log_2(\log_3(\log_4(\log_5x)))$?### Response: I want to find the set of values of $x$ that make the function $f(x)$ well-defined and real.
To do that, I need to consider the restrictions that each logarithm imposes on its argument.
Starting from the innermost logarithm, $\log_5x$, I know that its argument must be positive, otherwise the logarithm is undefined or complex.
So I have the inequality $x>0$.
Next, I look at $\log_4(\log_5x)$.
Its argument must also be positive, which means that $\log_5x>0$.
Using the properties of logarithms, I can rewrite this as $x>5^0=1$.
So I have a stronger inequality $x>1$.
Moving on to $\log_3(\log_4(\log_5x))$, I apply the same reasoning and get that $\log_4(\log_5x)>0$, or equivalently, $\log_5x>4^0=1$.
This implies that $x>5^1=5$.
Finally, I look at $\log_2(\log_3(\log_4(\log_5x)))$.
Again, I need its argument to be positive, so $\log_3(\log_4(\log_5x))>0$, or $\log_4(\log_5x)>3^0=1$.
This implies that $\log_5x>4^1=4$, or $x>5^4=625$.
Therefore, the domain of $f(x)$ is the set of all real numbers $x$ that satisfy $x>625$.