Why aren't logarithms defined for negative $x$? Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative numbers?
Why can't $b$ be negative? Take $(-2)^3 = -8$ for example. Turning that into a logarithm, we get $3 = \log_{(-2)}{(-8)}$ which is an invalid equation as $b$ and $x$ are not positive! Why is this?

In $\log_{−2}(8) = 1/2$, the way to solve it normally is to write: $(-2)^{1/2} = 8$.
But in this case, $(-2)^{1/2}$ is undefined. Hence, negative bases for log cannot be used.
