I have been witnessing alot of varying ideas on this subject - how high to mow grass. There are a few different viewpoints here, let me discuss with you;
1. A very short cut (mower cut ranges <1.75) - the resident may feel he or she will get the best value for the cut prescribed as he or she will not need you back again so soon... Maybe so on this regard, but dependent on the weather, grass type and soil type, you will find that the roots of grass will be exposed and often not be able to tolerate such heat contrast and will result in a quick die off of the lawn. It also exposes weeds to rear up where prickles become a nuisence, thus higher costs for poisons or the back break of weed picking, dead patches, etc. Recovery of the optimal lawn is generally harder once this has been done. It really only works on a golf course green, or alike, where much attention to maintaining the root stock - scarifying, fertilisations, rolling, watering program, etc.
2. A medium cut (mower cut ranges 1.75 - 2.25) - the cut will leave the lawn looking amazing (once again dependent on grass type), where the resident will feel the lawn has a bounce, manicured and able to cope with a dog or kids running or playing chase. The height of this cut allows the grass roots to still be protected, enabling a thickening of the grass. The weeds will find it hard to break through. The soil will remain nourished, cooled and protected as well, enabling the worms, bugs and other soil organisms to keep on moving and nourishing the soil. It down side is that it may become shabby quicker, best suited to a routinely mow.
3. A long cut (mower cut ranges > 2.25) - this cut is used if you have an unused paddock or a golfing piece of rough, or alike that may need a cut to keep it under control and pose as a fire hazard. The problem to watch is that the grass does not get that high that it tends to lay over, which then also acts alike the short cut, exposing the roots of some spots and other spots have a thick canopy. Weeds will generally find there growth through poorly nourished soils and heated exposure to sunlight. A higher cut may be beneficial if there is rock shale soils that expose hazards after rain, reducing the dirt and showing the emerging rocks. Also if there is guinea grass or other clumping rooted weeds.
Just a note, to go from a high cut lawn to a medium cut lawn, you would be encouraged to bring the depth of the cutting blades down slowly in increments over several weeks. This allows the long thin strands of grass to learn to thicken up from the change of conditions and maintain some protection of the root system.
Similarly, to go from a short cut to a medium cut, you will expect weeds to still be active until it has enough coverage to block out the weeds light and heat. The other thing that may help is to sprinkle cut grass (without heads of weeds) over the precut lawn allowing more canopy to the growing grass sections. The same is for growing grass - top soil is a must for any accererated growth of grass and grass crawling strands or headed seeds strategically positioned. And of course, some light regular watering and fertisation.