Logarithmic fever

$\begin{aligned} \log_\frac 34 \left(\log_8 {\color{#3D99F6}(x^2+7)} \right) + \log_\frac 12 \left(\log_\frac 14 {\color{#3D99F6}(x^2+7)}^{-1} \right) & = -2 & \small \color{#3D99F6} \text{Let }u = x^2 + 7 \\ \log_\frac 34 \left(\log_8 {\color{#3D99F6}u} \right) + \log_\frac 12 \left(\log_\frac 14 {\color{#3D99F6}u}^{-1} \right) & = -2 \\ \log_\frac 34 \left(\frac {\log u}{\log 8} \right) + \log_\frac 12 \left(- \frac {\log u}{-\log 4} \right) & = -2 \\ \log_\frac 34 \left(\frac {\log u}{3\log 2} \right) + \log_\frac 12 \left(\frac {\log u}{2\log 2} \right) & = -2 & \small \color{#3D99F6} \text{Let }y = \frac {\log u}{\log 2} \\ \frac {\log y - \log 3}{\log 3 - \log 4} + \frac {\log y - \log 2}{\log 1 - \log 2} & = - 2 \\ \frac {\log y - \log 3}{\log 3 - \log 4} - \frac {\log y}{\log 2} + 1 & = - 2 \\ \log 2 (\log y - \log 3) - \log y (\log 3 - \log 4) & = -3 \log 2 \log 3 - \log 4) \\ (\log 2 - \log 3 + \log 4)\log y & = -3 \log 2 \log 3 + 6\log^2 2 + \log 2 \log 3 \\ (3\log 2 - \log 3)\log y & = 2 \log 2(3\log 2 - \log 3) \\ \log y & = 2 \log 2 \\ \implies y & = 4 \\ \frac {\log u}{\log 2} & = 4 \\ \log u & = 4 \log 2 \\ \implies u & = 16 \\ x^2 + 7 & = 16 \\ \implies x & = \boxed{3} \quad \small \color{#3D99F6} \text{for }x > 0 \end{aligned}$