I see why you might be confused. First, the logistic-loss or log-loss is technically called cross-entropy loss. This function is very simple:
$CE = -[y \log(p) + (1 - y) \log(1 - p)]$
This tells basically if the predicted class $y$ was right $y=1$ then the loss is $CE=-\log(p)$, if the predicted class was not the right one then the loss is $CE=-\log(1-p)$.
If we look at the function as a pure math concept we see that:
$CE = f(x) = - \log(x)$
And as you point out, that function is minima-unbounded as its domain is $D(f(x)) = [0, +\infty]$. You can check that in here:
However the trick is that the inputs must be bounded, meaning, the inputs to the loss function must be in range $[0, 1]$. This bounding is achieved by applying a sigmoid activation function as the final "layer" of the network. Then if the inputs to the loss function are bounded the function has a clear minima.
Check how the function looks in reality from one of the most important papers in loss function in the AI world: Focal Loss (I really encourage you to read it as the first section explains in detail the cross-entropy loss). The blue curve is the one you are looking for.
Finally, you might want to review your log-loss/CE function since it should have an asymptote for $f(x=0) = \infty$