No problem.
Just log your hours, using the same logbook. Nobody will care or ask. Don't go about calling LT with these questions, they don't care about your well-being or your (lack of) experience. Go ahead, log your flying hours as you gather experience, use a single logbook. No worries. I've done so over the last 15 years, flying everything from the C152 to the B757, US and European aircraft. No problem, I'm still alive, earning my living from flying and various European CAA inspectors have been digging into my documents on a number of occations (SAFA ramp checks) without as much as a comment. Don't worry too much about these silly details you're asking about.
Again, there's only been ONE case in which LT has scrutinized logbook(s) - and that was a case which was just too obvious, calling for LT action.
Happy landings.