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Relevant wiki: Real Numbers

$\begin{aligned} \frac {8^x+27^x}{12^x+18^x} & = \frac 76 \\ \frac {2^{3x}+3^{3x}}{2^{2x}3^x+2^x3^{2x}} & = \frac 76 \\ \frac {(2^x+3^x)(2^{2x} -2^x3^x+3^{2x})}{2^x3^x(2^x+3^x)} & = \frac 76 \\ \frac {2^{2x} -2^x3^x+3^{2x}}{2^x3^x} & = \frac 76 \\ \left(\frac 23\right)^x - 1 + \left(\frac 32 \right)^x & = \frac 76 & \small \color{#3D99F6} \text{Let } y = \left(\frac 23\right)^x \\ y - 1 - \frac 1y & = \frac 76 & \small \color{#3D99F6} \text{Multiplying both sides with } 6y \\ 6y^2 - 6y + 6 & = 7y \\ 6y^2 - 13y + 6 & = 0 \\ (3y-2)(2y-3) & = 0 \end{aligned}$

$\implies y =\left(\dfrac 23\right)^x = \begin{cases} \dfrac 23 \\ \dfrac 32 \end{cases} = \left(\dfrac 23\right)^{\pm 1} \implies x = \boxed{\pm 1}$