The Math Hack That Only Works by Accident

Ajouté : 2026-05-22

[00:00:00]I saw this meme and I wanted to explain what it means, which pro tip is the absolute best thing you can ever do to a joke. So, one of the first things we learn when we start learning about integers and addition and multiplication is that multiplication distributes over addition, meaning it's kind of like a hack around order of operations. If I want here, I can of course start inside the parentheses, add the two and the five, which gets me seven, and then multiply three times seven, which of course gets me the result of 21. Or if I prefer, I can actually just do the multiplication here first, multiplying three times two to get six, and then three times five to get 15, and then adding the six and the 15 together to get 21. So, either way I do it, it doesn't matter, I still get 21. And this is great, this is fantastic, right? The only problem is multiplication over addition is one of the only times that distribution works. It's very naturally when you start learning higher mathematics to presume, "Hey, I bet lots of things distribute. Maybe when I take the sign of some angle plus another angle, it's equal to the signs of the individual angles added together." But unfortunately, it's just not true.

[00:01:08]Similarly with the logarithm, that natural logarithm does not distribute.

[00:01:12]It is not true in general that the natural logarithm of A plus B is equal to the natural log of A plus the natural log of B. In fact, if it were, it would kind of defeat the whole purpose of logarithms in the first place. What makes logarithms so amazing is that they turn multiplication into addition. The whole point here is multiplication is hard, but addition is easy, and the logarithm is what lets us turn multiplication into addition. So, this is like a fundamental property of logarithms, which is why it can be so strange and surprising and dare I say even frustrating when the student who is first learning logarithms sees this crazy result. Natural log of 1 + 2 + 3 is in fact equal to the natural log of 1 plus the natural log of 2 plus the natural log of 3. Why would this be?

[00:02:02]Well, it's not that natural log suddenly distributes for some reason. It's just that we have a funny coincidence based on this property we stated about logarithms a moment ago. The logarithm property tells us if we're adding ln1 plus ln2 plus ln3, that should actually be the same as the natural log of 1 * 2 * 3. And just by coincidence, it turns out that 1 + 2 + 3 happens to be the same thing as 1 * 2 * 3. Of course, what I'm talking about here is 6. 1 + 2 + 3 is 6 just like 1 * 2 * 3 is 6. And so, in this particular case only, not generally, don't upset your 10th or 11th grade algebra 2 teachers, but in this particular case, it is true that we end up with something that looks like distribution. The natural log of the quantity 1 + 2 + 3 is in fact equal to the natural log of 1 plus the natural log of 2 plus the natural log of 3.