Logarithms and Logarithms!

$\begin{aligned} \log_x \frac 18 & = - \frac 32 \\ x^{-\frac 32} & = \frac 18 = 2^{-3} \\ \left(x^{-\frac 32}\right)^{-\frac 23} & = \left(2^{-3}\right)^{-\frac 23} \\ \implies x & = 2^2 = \boxed{4} \end{aligned}$

$\begin{aligned} \large \log_x \dfrac 18 & = -\dfrac 32 \\ x^{-\frac{3}{2}} & = \dfrac 18 \\ x & = \sqrt[\frac{-3}{2}]{\dfrac 18} \\ x & = \left(\dfrac 18\right)^{\frac 1{-\frac 32}} \\ x & = \left(\dfrac 18 \right)^{-\frac 23}\\ x &= \dfrac {1}{(\frac18)^{\frac 23}} \\ x &= 8^{\frac 23} \\ x & = (2^3)^{\frac 23}\\x &= 2^{\frac 63}\\ x &= 2^2 \\ \end{aligned}$

$\large \implies x = \boxed{4}$