Hi Coppit
I shoot the moon often, both with and without a tripod. I do not have an LX3, but I shoot regularly with the FZ 50, FZ 35, and an Olympus DSLR. I use the same settings on all, and am very pleased with the output.
Set the ISO as low as it goes (80 to 100). Spot meter, and Spot focus. F stop between 5.6 and 7.3 and the shutter speed between 500th and 800th of a second. You can experiment with which of the f stops go best with which shutter speed. I typically stay at f 5.6 and 600th of a second.
If you use a tripod, with all the above settings, you should try shutting off the OIS. and use the 2 second timer to help reduce shake. This should give you the best shot, but if you brace yourself against something, hand held comes out nicely as well. You can also try the 2 second time when hand held, which seems to help a smidge....
The moon is very bright, so that is why these higher speeds work better, plus the F stop above 5.0 minimizes any fringing....
Give it a whirl, and let us know how the LX3 does....
Larry
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Olympus E-510 DSLR with 14-42 mm & 40-150 mm Lens
Lumix FZ 35, FZ 50, FZ 20, TZ5, TZ3, FX 07, TCON 17, MCON 35, RDS, Sunpak 383.
Fuji F-20.