Give the region of validity. Let $F(z)=\log(z^3-8)$.
Let $z=re^{i\theta} \Rightarrow z^3=r^3e^{i3\theta}=8 \Rightarrow r=2$ and $\theta=\frac{2\pi k}{3}$ for $k=0,1,2,3$. So, $F$ is analytic on $\mathbb{C}$\{$z=re^{i\theta}:r>2,\frac{2\pi k}{3}$ for $k=0,1,2,3$}. Here we can take our branch cut to be $s:=${$x+iy:x>2,y=0$} 
Is this correct? Can someone just explain what exactly we are doing when we make a branch cut? 
 A: Your cut(s) is(/are) indeed correct. I'll use this post mostly to talk about what a branch cut is and why we do it.
When we write $\textrm{log}$ as a power series, it has a radius of convergence - it converges in a circle that goes out to $0$. We can analytically continue this. However, unlike many functions you see in complex analysis, you can't extend this perfectly to the whole plane. If we had picked a point in this radius of convergence and drawn a new circle around it, we'd have more regions for which our function was defined (and holomorphic in). But if we kept making circles like this all the way around the singularity at $0$, and got back where we started, we wouldn't have the same value we had before - it would have increased (or decreased) by $2\pi i$. This is trouble - we want $\ function to take on a single value!
There's two things that, when combined, cause this problem. First, there are many different values of $z$ that satisfy $e^z = z_0$ (for fixed $z_0$). Secondly, if you remove a point from the complex plane, the space you get fails to be what's called "simply connected", which means that if we draw a loop in it, we can't necessarily squish that loop down to a point - in particular, any loop you draw around $0$ (the point we removed) can't get squished down. To describe why these two fit together to cause a problem, let's pretend our loop is just the unit circle. Then $\text{log}(e^{i\theta})$ keeps increasing as we go around the circle - if it started at $0$, it ends up being $2\pi i$. If we had been able to squish the circle down to a point, then by continuity, the values along the circle all get close to each other - in particular, when we squish it down to a point, they'll all be the same (including when we trace that path out) - so there's no problem with the function being multi-valued.
But we can't squish the circle. So what we do to solve this is we make a "branch cut" - we make the space simply connected. If we delete the negative real axis, then the space we get is again simply connected; the original thing that stopped us from pulling the loop to a point is that we had a point "inside of it" that we deleted - but good luck drawing a loop in the new space that has one of these deleted points inside. We give $\textrm{log}$ a canonical value in this new space (usually $\text{log}(1)=0$, though in yours you'd probably use $\text{log}(i):=\pi i$), and since we can't get multiple values by drawing those non-squishable loops, our function has a single value at every point it's defined, is holomorphic, and pretty much every property we could want. 
In summary: a branch cut stops our function from taking on multiple values by making it so we can't draw loops around the singularity. We do it because nobody wants a function whose value can be an infinite number of different things!
