Connecting gas logs to small propane tank
I just did the same thing. I installed an Empire vent free log set in my wood burning fireplace that I plan to use mostly in a vented manner.  I ran black pipe from the outside on my open air back porch into the fireplace which attached to the gas log burner controls.  

For the outside, I simply attached a standard BBQ replacement propane hose with regulator to a 20lb tank.  It works just fine.  My log set is around 32000 btu so I can expect around 10.5-11 hours on a 20lb tank.  I plan to get a 40lb tank.

One issue for you might be if your logs don't have the proper flame, it may be due to the hose used to connect to the tank being 1/4".  Try to find a hose/regulator that is 3/8" which should provide a bit more gas to the logs.

Be aware that local code might require you to have a gas shutoff installed inside the house.  Mine does, so I installed one.   

One caution.  Be sure to get a gas log set that does NOT have a continuous pilot light.  A pilot light running 24/7 will use about 8 gallons of propane a month.  That's 2 20lb tanks worth of propane.