Logs and Logs!

Let $y = \log_{ac} x$ .

Using the logarithmic property, $\log_p q = \frac1{\log_q p}$ , we can rewrite all the given equations as $\begin{cases} \log_ x ab = \frac19 \\ \log_x bc = \frac1{18} \\ \log_x ac = \frac1y \\ \log_x abc = \frac18. \end{cases}$

And because $\log_p q + \log_p r = \log_p qr$ , the sum of the first 3 given equations is equal to twice the remaining given equation, $\log_ x ab + \log_x bc + \log_x ac = 2 \log_x abc \quad \Leftrightarrow \quad \frac19 + \frac1{18} + \frac1y = \frac28 \quad \Leftrightarrow \quad y = \boxed{12} .$

$\begin{cases} \log_{ab} x = 9 & \implies x = (ab)^9 & \implies ab = x^{\frac 19} & ...(1) \\ \log_{bc} x = 18 & \implies x = (bc)^{18} & \implies bc = x^{\frac 1{18}} & ...(2) \\ \log_{abc} x = 8 & \implies x = (abc)^8 & \implies abc = x^{\frac 18} & ...(3) \end{cases}$

$\implies \begin{cases} \dfrac {(3)}{(2)}: & a = x^\frac 5{72} \\ \dfrac {(3)}{(1)}: & c = x^\frac 1{72} \end{cases} \implies ac = x^\frac 1{12} \implies x = (ac)^{12} \implies \log_{ac} x = \boxed{12}$

I hope this will help.