Exponential Equations

Solve for $x$: $8^{2x-1} = \left(\frac{1}{2}\right)^{x+3}$.

The populations of two bacterial cultures are given by $P_1 = 9^{x^2 - 1}$ and $P_2 = 27^{2x + 1}$, where $x$ is time in hours. At what time $x$ (in hours) are the populations equal? Solve for $x$.

Solve for $x$: $2^{x^2 - 2x} = 4^{x+3} \cdot 8$.