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 $\log_c{x}$.

 A: 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:
$$
\log_c x = \dfrac{\log x}{\log c} = \boxed{ \dfrac{1}{\frac{1}{r} - \frac{1}{q} - \frac{1}{p}}}
$$
A: You can form these expressions:
$$\log_c x = \frac{\log_a x}{\log_a c} = \frac{\log_b x}{\log_b c} = \frac{\log_{abc} x}{\log_{abc} c}$$
These are four unknowns ($\log_c x$ and $\log_\ast c$) and three equations, so we seem to be a bit lost, since it seems hard to relate the $\log_\ast c$ to each other.
Do you have additional information on $a,b,c$? Constraints or an equality or even values?
EDIT
Adriano has found the required additional equation. Confer his answer for the full solution.
