You can ballpark Decibels by remembering two values... 3 is either double or half (depending on whether it's gain or loss) and 10 dB is 10 times (or 1/10th) the original value.
In the example given, the 12 dB of gain in the antenna is reduced 6 dB by the loss in the feed line. Overall antenna system gain is now 6 dB which is 3dB + 3dB or double, then double again, for a multiplier of 4. If we want 500 watts ERP we'd need 125 watts from the transmitter. The first 3 dB of gain takes the ERP to 250 watts, the second 3 dB of gain takes the ERP to 500 watts. Or, working the reverse, 500 divided by 4 = 125.
Here's an example for you to work: An amplifier is rated at 23 dB of gain. What's the ratio between the input and output level?
This really helps. So from your example I divided 23 by 3 and came up with 7.6 (approx). Then I used an arbitrary output of 5 watts power. Doubling that 7.6 times would be 5X27.6
which gives me about 970 Watts. Correct?