1. The problem is to solve the equation $8^z = -7$ for $z$. 2. Recall that for any real number base $a > 0$ and $a \neq 1$, the exponential function $a^x$ is always positive. This means $8^z > 0$ for all real $z$. 3. Since the right side of the equation is $-7$, which is negative, there is no real solution to $8^z = -7$. 4. If we consider complex numbers, the solution involves complex logarithms, but typically in algebra, we restrict to real numbers. 5. Therefore, the equation $8^z = -7$ has \textbf{no real solution}. Final answer: \text{No real solution for } z \text{ since } 8^z > 0 \text{ for all real } z.