<s>[INST] Find $\log_c{x}$ if $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$. 
Given that $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$, find the value of $\log_c{x}$.

 [/INST] Using the change of base and product rule for logs, we have:
$$
p = \frac{\log x}{\log a}
\qquad\text{and}\qquad
q = \frac{\log x}{\log b}
\qquad\text{and}\qquad
r = \frac{\log x}{\log abc} = \frac{\log x}{\log a + \log b + \log c}
$$
Taking reciprocals of each equation, we can combine them to obtain:
\begin{align*}
\frac{1}{r} - \frac{1}{q} - \frac{1}{p}
= \frac{\log a + \log b + \log c}{\log x} - \frac{\log b}{\log x} - \frac{\log a}{\log x}
= \frac{\log c}{\log x}
\end{align*}
Taking reciprocals again, we conclude that:
$$
\log_c x = \dfrac{\log x}{\log c} = \boxed{ \dfrac{1}{\frac{1}{r} - \frac{1}{q} - \frac{1}{p}}}
$$
</s>