When my 11 old son was born, I told a friend that I have this funny goal to teach him how to mentally calculate natural logarithms by the time he's in 4th grade

When my 11 old son was born, I told a friend that I have this funny goal to teach him how to mentally calculate natural logarithms by the time he's in 4th grade. Time flies, and before I knew it he was in the last week of the 4th grade. So, I taught him just that. Mechanically. He is now in 6th grade (first day of middle school yesterday, big day). He forgot how to do these logarithms, and he never really understood them, obviously. But he found the trick entertaining, and I hope when he'll get to learn them in high school he'll find them as old friends. I don't know, we'll see. In any case, calculating natural logs with 2 decimals accuracy is a cool nerdy trick to do. Everyone has a smart phone, you can ask them to start the calculator app, run it in extended mode (tilt the phone horizontally), and press the rand button. For example, right now I got 0.874. Announce that you'll calculate the natural log with 2 exact decimals or thereabouts. You start with 0.9, which has log(0.9) = log(9/10) = log(9)-log(10) = 2log(3) - log(10), and you have memorized a short table of logs of the integers between 1 and 10, and know that log(3) = 1.1 and log(10) = 2.3, so what you have so far is -0.1 . Then you say notice that 8.74 = 9-0.26, you are roughly 2.6% lower than 9. More precisely 2.6% divided by 9, which is very close to 2.6% multiplied by 1.1, and you know the multiplication by 11 trick (add the digits and put the result in the middle), so all in all 8.74 is about 2.9% less than 9. you subtract this number from the logarithm, but first you round it to a whole percentage, so you subtract 3% from -10% and you end up with -0.13. The actual value is -0.1347. In other words, you use two formulas, log(ab) = log(a) + log(b), log(1+x) = x for small x (this is the first term in the Taylor series, and for our purposes it is exact if x is a single digit percentage). Kids can learn these manipulations mechanically at an early age, just like they can learn simple magical tricks. And they are having fun with it. Later on, they don't get confused by the formula log(ab) = log(a)+log(b) because they have used it and are familiar with it. PS. By the way, I once heard the apocryphal story that Gauss was able to calculate any natural logarithm in his head with 5 exact decimals. I did not find (or look for) a confirmation in the literature, but I find it believable. If you teach your kid the trick with the log up to 2 decimals, and they already learned to do the sum of the numbers up to 100, you can tell them they are like a mini-Gauss.