That is the same logo configuration that was on my 1947 L-7. Mine had a white oval label with A-prefix serial number (visible inside the bass-side F-hole). That label came in during April of 1947, the best I can tell. Guitars prior to that probably would have a factory number (FON) ink-stamped on the inside back, which should be visible through one of the F-holes if you look closely with a small flashlight.
If there is no factory order number, look for evidence of a lost label, such as an oval glue pattern on the inside back.
The inside of the guitar is probably clear-coated rather than bare wood.
The original tuners, if you don't have them, would be individual closed-back single-ring Klusons with plastic keystone buttons. Those are still readily available in modern repro versions.
One interesting detail is that there appears to be a screw hole for a finger rest (pickguard) bracket on the bass rim, near the waist. That is typically seen on the opposite rim on a right-handed guitar. It is possible the guitar was set up as a lefty at some point in its life.
In any case, as Jim says, this appears to be an L-7 from no later than 1947. By 1948, this would have had the modern block logo instead of the script logo.
If you an find an FON inside, we can pin it down more closely.
If you need a case for it, any case fitting a J-200 or other 17" Gibson flat-top should fit, since these archtops have the same body shape as the J-200. The L-7 is probably slightly thinner, but because of the carved top and carved back, it takes up the same room in the case as a thicker-bodied flat top.