I see quite often people doing a lot of sit-ups for core (or just to look nice in summer); but not a lower back as well. Core is defined as our abdominals, lower back, buttocks, and hips, and all the core muscles must act together to stabilize the spin.
I am doing (when in mood) two ways of strenghten the core; repeadetly flex, extend, and rotation the trunk (like crunch, psoas crunch, swimming on dry up to 30-45 seconds), and onother method to stabilized the trunk against the force of gravity (isometric, static strenght) something like front plank, side plank, push up position etc. up to 30 sec or 1 minute.
Important is to focus on all the core muscles. Many people has issues with lower back pain as result of "strong" abdominal and "weak" lower back, strenghtening just one group is not gonna help a lot.
Hope this help!