Solve the equation: $\;$ $\log_3 \left[1 + \log_3 \left(2^x - 7\right)\right] = 1$
Given equation: $\;\;$ $\log_3 \left[1 + \log_3 \left(2^x - 7\right)\right] = 1$
i.e. $\;$ $1 + \log_3 \left(2^x - 7\right) = 3^1 = 3$
i.e. $\;$ $\log_3 \left(2^x - 7\right) = 2$
i.e. $\;$ $2^x - 7 = 3^2 = 9$
i.e. $\;$ $2^x = 16 = 2^4$
$\implies$ $x = 4$
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{4 \right\}$