Much will depend on the generator type and starting type on log splitter.
It would continue to run of course, but the main question is will it start? Many generators will allow overload for few seconds to start large motors but with some the field fails when overloaded.
The modern softstart and inverter drives start motors much slower than the of capacitor or star - delta start and using one of these may mean a motor which would not normally start can now be started.
Motors which start no load can often be switched on and off a few times to let it gather speed.
The high start up current in the motor before it reaches full ( or normal operating ) speed may be too much for the generator and it may trip or fold back ( reduce its ouput voltage to prevent overloading ). If it trips then a soft motor starter will be needed. If it folds back then the motor may never reach full speed.
A simple ( or bodged ) soft start can be a 1 kilo watt fire bar element in series with the motor ( if motor 3 kilowatt ) when starting the motor and then shorted out of circuit when the motor has reached a good speed.
One thing to be careful about if the generator does start the motor is how the generator reacts to the motor's startup load. If the generator has a sudden change of speed this might be putting a strain on the mechanics of the generator and engine.
And if anything else is powered by the generator then that equipment is likely to get a short but possibly destructive over voltage when the motor is switched off.
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