A logical qubit is a very fluid concept. You could use physical qubits as logical qubits. Or, you can encode multiple physical qubits as a single logical qubit. The more physical qubits you use, the better the resistance to noise. So, I would suggest that you question isn't exactly the right one to ask, and a better question is whether something useful can be done with existing quantum technology (in the direction of computation).
The long-term goal is to build quantum computers, which require logical operations to be performed with a suitable level of reliability (below the "fault-tolerant threshold"), that can be maintained for a long time. It's true that we're probably not quite there yet, even with a single logical qubit. I don't actually know how close current hardware is to achieving it. That doesn't mean that existing devices are entirely pointless.
There is a lot of research going on at the moment into "quantum supremacy", in other words, given the sort of noisy quantum devices of 50-100 qubits that are starting to appear, is there anything that we could do with them that is unequivocally better than anything we could do with a classical computer? The expectation is that we're somewhere around that threshold at the moment, but I'm not aware of anything that is definitive.