The relation I quoted is you can also find in "Radar Encyclopedia" that can be found here in download section. The values are all in natural dimensions i.e. Watts, not dBW or dBms. If your power is in dBs than the relation is Pa=Pmax+10*log(duty_cycle). It is the same like in your expression log(1/x) = -log(x)
For my sense, involving antena gain in the way the expert said means that the peak power is radiated (EIRP) and average power is the power at the antenna port. The relation can be applied if you want to determine average power that that amplifier delivers to the antenna if you now the peak EIRP (it is for me uncommon).
Maybe it helped
Regards