Exactly, (again) Chickenhouse has got it as have some other learned folk here - - as chickenhouse and porterhouse have correctly declared without the academia thereof
Load your 172 or PA28 or 152, A320, A330, B747, DC10 chocka block with weight - up to the max perameters etc - or not, just for this academic experiment and try to fly as you would if you were with full tanks and no payload and no other baggage, with same power setting
~and
You would notice a decrease in airspeed in the heavier aircraft. Assuming then that you could actually fly the heavier aircraft at that power setting (and speed) - assuming you could control the aircraft and that it did not feel too sloshy you would be grappling for speed, never mind the handling.
Now, if you apply full power AND hope for the best, in this experiment, for we are experimenting here guys (and fair maidens
then, you might, might get an increase in airspeed and with a lower angle of attack, (providing you are not too heavy, baby) and hopefully the handling characteristics would be perhaps tighter affording greater flying control of the aircraft. [if not then land the bloody thing and consider you got away with it]
In short -
For a substantial increase in weight - you need an increase in power.
The manuals of all of the above aircraft give you the graphs and the tables so you can all go and work it out for yourself before you fly - which is what you do.
But, as we all train in an aeroplane with no payload apart from the instructor or flight crew but no pax or payload - and we already know what is written in the performance tables - and even though we know that weight is opposite to lift and the less weight you have the more lift you get - in short.
No one stands there in front of you all in class, and announces that
for an (a substantial) increase in weight you WILL need an (a substantial) increase in power.
Nor is it writ, in them there books like you say on the first pages either. Obvious - simple and true.