Why are logarithms not defined for 0 and negatives? I can raise $0$ to the power of one, and I would get $0$. Also $-1$ to the power of $3$ would give me $-1$.
I think only some logarithms (e.g log to the base $10$) aren't defined for $0$ and negative numbers, is that right?
I'm confused because on all the websites I've seen they say "logs are not defined for $0$ and negative number". On one website it says "$\log_b(0)$ is not defined", then provides an example where the base is $10$.

You can define everything you want, but will this newborn object satisfy properties you want, depends on your definition. Assume, we do have logarithms for negative numbers and zero and all the properties of logarithms are preserved. Then we immediately obtain a contradiction. Here it is
$$
0=\log 1=\log(-1)^2=2\log (-1)
$$
so $\log(-1)=0$ and from the definition of logarithms we have $-1=10^0=1$. This is one of the reasons. 
But if you still want to take logarithms of negative numbers, you must relax some requiremetns. The most reasonable is to make logarithms multivalued with values in $\mathbb{C}$. For more detailed description of such logarithms look  at Complex logarithm
