I had a physics prof from England that would walk us through electromagnetics problems and when he'd get to the end it was a mess of an expression and he'd say " ... which resolves to .... approximately 37.57" and we'd follow along with out calculators and the answer would 37.55.
We asked him how this was possible. he said 'I convert it all to logarithms in my head and then it's just simple addition. Haven't you memorized all the basic logs for all the major constants like Pi, e, u0, boltzmans, etc and the digits 1-10?"
THAT guy was a rock star to the math nerds.
THAT is serious business!
I remember a “Math Magic” article in Popular Science Magazine around 1962 or 1963. It addressed how to mentally simplify problems to get to quick answers.
In engineering school, we would use logs like you described to quickly get to an approximate answer. That way you knew if your final solution made sense. We also mastered “dimensional analysis” for dealing with the units on all sorts of variables and constants. If you couldn’t get the units to work out, you were off base to start with.
I’ve taken up lawn care as a hobby and I’m appalled at the huge mishmash of units for chemicals to be applied to the lawn. Pints of Product/Acre, Pints of Active Ingredient (AI)/Acre, Grams AI/Hectare, Ounces/1,000 Sq Ft (abbreviated “M”) and lots more. I’ve had to make up a spreadsheet to normalize all the different units on all the products I’m using. I eventually get down to how many ounces of product I need to add to X gallons of water. It’s not rocket science, but all those tools I learned years ago come in handy. While it’s not rocket science, you can easily screw up and burn up your lawn.