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How I Feel About Logarithms
I like the number 8. I like the way it smells, like 2 and 4, with a hint of 3, in a cubic sort of way. Though that only serves to further support the sweet scent of 2. There is a need, deep within me, to express how I feel about logarithms. But I worry they've already been ruined for you, obscured by the shuffling of numbers. This shuffling, shuffling, soulless zombie step, suffering the mind's logarithmic decay. Institutions full of innocence drawn into that dance of death. And yet, if there's one thing with true universal appeal that all humans share, it is not love, not desire for food, shelter, or safety. It's mathematical truth, which applies equally to us all, the normal and the abnormal, the living and the dead. That so much of school is wasted on nitpicky particulars of elementary algebra is incredibly tragic. Elementary algebra is just fancy counting, all of it. You start by learning to count. You were taught a sequence of syllables. And you parroted it back for the reward of some authority figure telling you, "good job," or sometimes, "very good." And then, you were told that 1 plus 1 is 2. And parroted that back, too, playing a call-and-response game of nonsense syllables until eventually you figured out the fundamental idea of algebra. There is this ordered sequence of things, these numbers. And they have one true behavior, plus 1. Each individual symbol or syllable is just a shorthand for some plus 1's. 2 is positive 1 plus 1. 3 is plus 1 plus 1 plus 1. When we say 2 plus 3 equals 5, it's not that adding 2 and 3 results in 5 or that 2 plus 3 is different but equivalent to 5. But that 2 plus 3 is 5. They are different names for the same exact thing, which is plus 1 plus 1 plus 1 plus 1 plus 1. That's it. Plus 1 is all there is. And no matter how you group those plus 1's, all that matters in the end is that you don't lose any. Oh, then, there's reverse plus 1's, negative numbers, subtraction, counting down. Those are just plus 1's again, but going back in time. 7 minus 5 is just counting to 7, plus 1 plus 1 plus 1 plus 1 plus 1 plus 1 plus 1. And then, going back in time for 5 plus 1's. You can think of it as mutually annihilating plus 1's and anti-1's and seeing what's left over, if you want. 8 minus 2 is just counting to 8, and then, going back in time 2 counts. 1 minus 4 is when you find that counting from 4 to 1 takes negative time. You're done 3 counts before you started. Or the 1 annihilates with one of the four anti-1's and three anti-1's are left. And if you start with two negative numbers, you're already back in time and only go further back. You've just got a pile of anti-plus 1's trapped in the past. Subtraction is egocentric counting. 7 minus 4 means you are 4. You are the center of the universe and compared to you, what is 7? Well, just count to it, plus 1 plus 1 plus 1. From where you're looking, 5 is plus 1. 3 is also just plus 1 away, but against the flow of time. So it's an anti-1. Minus 4 is twice as far away back in time as it would be if you were looking from a 0 perspective. To get to negative 8, sometimes to make the harder things simple, first, you have to make the simple things harder. Addition is a process of plus 1 plus 1 plus 1. Multiplication is a process of plus n plus n plus n. For multiples of 2, instead of counting plus 1 plus 1 plus 1, you're counting by plus 1 plus 1, plus 1 plus 1, or plus 2, for short. Plus 2 plus 2 plus 2 plus 2. Fancy counting. Counting in a plus 2 kind of way. Counting in a plus 5 kind of way. Counting in a plus 10 kind of way. What is 5 times 7? All that means is count in a plus 5 sort of way seven times, which happens to be the same as counting in a plus 7 sort of way 5 times. 1, 2, 3, 4, 5. This is why 8 smells like 2 and 4. Because 8 is 4 when you're counting in a plus 2 sort of way. And 8 is 2 if you're counting in a plus 4 sort of way. Division is fancy counting. A divided by B asks the question, what is A, when counting in a plus B sort of way? 27 divided by 9 means, what is 27, in a plus 9 sort of way? Plus 9 plus 9 plus 9, it's 3. What is seven in a plus 2 sort of way? Just count, 1, 2, 3, and 1/2. Division means counting by plus denominators until you get to the numerator and seeing how long it takes. 7 divided by 1/3 means counting by plus 1/3's until you get to seven. Plus 1/3 plus 1/3 plus 1/3 plus 1/3 plus 1/3, all the way to 21. Exponentiation-- powers-- also fancy counting. This time, instead of counting by plus whatever, you count by times whatever. Powers of 2 are just counting in a times 2 sort of way. First power of 2, second power of 2. third power of 2. The size of the times-ish step is the bottom number. And the number of times-ish steps is a little exponent. Powers of 10 count in a times 10 sort of way. To take 10 to the sixth, you just take six steps of size times 10, 10 times 10 times 10 times 10 times 10 times 10. That's why 8 has a whiff of 3. 8 is 3 when counting in a times 2-ish way. Then, there's roots. To take the fourth root of 81, you're just saying, hey, if 81 is 4 in a timesy sort of way, then how are we counting? It's like going a quarter of the way there, but a timesy quarter, not an addish quarter, times 3 times 3 times 3 times 3. As far as times-ish counting is concerned, 3 is a quarter of 81, which is why 81 to the 1/4 is the same as the fourth root of 81. And personally, I think it's kind of weird that we keep around this root notation, when fractional powers are so much more descriptive. 81 to this 3/4 means you're going 3/4 of the way to 81, in a times-ish counting way. A times-ish 3/4 of 81, simple. Writing that as fourth root 81 all to the power of 3 is just, why would you do that? Anyway, roots or factional powers are the case where you know how many steps you want to go, and what you want to reach when you get there. Six times-ish steps to get to a million means x to the sixth equals 1 million. The base is the way the exponent counts. And the way to count to a million in six steps is by times 10's. But say, you know the size of your step, and you know what number you want to get to, but you don't know how many steps to take. If you're counting in a times 3 sort of way, how many steps to 81? Or let's write that in a way that solves for x. 3 to the something is 81. So in a system that counts in a times 3 sort of way, 81 equals four steps. When I read the notation, I visualize it. STCx 2 of 8. System That Counts in a times 2 sort of way. That's a line of times 2's going on forever. Then, the 8 is somewhere in there. And how do I get there? In a system that counts in a times 2 sort of way, 8 is three steps. For a system that counts in a times 4 sort of way, 16 is 2. STCx 7 of 343 is 3. STCx 2 of 1/4 is, well, in a system that counts in a times 2 sort of way, now, you're going back in time, dividing by 2. Once, twice, that's negative 2 steps. The STCx 125 of 25 means we have a system that counts in a times 125 way. But the number we're trying to get to is only a fraction of the way to 125. And we can divide this step into three equal fractional steps, which times up to 125, each of size times 5. Not plus 5, as if we were dividing the normal way. But the times 5 of dividing in a times-ish way. Times 5, times 5, that's 25. Two steps along. Just one more times 5 to 125, making 25, 2/3 of 125, in a timesy counting way. In a STCx 64, 128 is not a whole multiple of 64, but a fraction past it. Luckily, you can divide 64 into 6 equal times 2 parts. Times 2 times 2 times 2 times 2 times 2 times 2. And just add one more times 2 to get to 128. Can you feel the 7/6-ness of 128 in relation to 64? Can you sense how 64 and 128 together evoke 32, which is 5/6 of 64? And 5/7 of 128 on this scale. Can you smell the sweet scent of times 2 times 2 times 2 to the fifth, 2 to the sixth, 2 to the seventh? You can almost taste 256. This, my good friend, is the logarithmic scale. This, my dear future inventor of the mortal space robots, is logarithms, a beast which at times transcends even the fanciest of counting. And as you plus 1 your way towards your noble goal, I only hope that you allow yourself the occasional moment to stop and smell the factors.