Find $\log_c{x}$ if $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$.

Question

Given that $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$, find the value of $\log_c{x}$.

Answer 1

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: $$ \log_c x = \dfrac{\log x}{\log c} = \boxed{ \dfrac{1}{\frac{1}{r} - \frac{1}{q} - \frac{1}{p}}} $$

Answer 2

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.