eli5: What exactly is a logarithm?
In: Mathematics
The logarithm of a number is easiest to think about as “how many digits do I have to write down to write this number?”. Think about the number 1 and the number 1000. 1000 is way bigger than 1 (a thousand times bigger), but it only takes four times as many digits to write down.
Because we use a number system where we have more than one digit (0-9), we can write big numbers without using a big number of digits. Imagine if we only had one number: 1. Then to write 5, we would have to write 11111. To write 10, we would have to write 1111111111. Writing 1000 would be absurd.
But because we can use more symbols, we can write big numbers in pretty small ways. In our case, we use 10 symbols (0-9) so we are in “base 10”. The logarithm of a number, base 10, is roughly the number of digits needed to write it down.
This isn’t exact – it’s only exactly true for numbers like 1, 10, 100, 1000, 10,000, and so forth. For the other numbers, the logarithm (base 10) is some number between the integers.
And we can use other bases. There’s binary (base 2), octal (base 8), and any other number of bases we could use. We can even use bases that aren’t integers. So as you go deeper the math will get a little more funky, but it’s always the same basic idea: how many digits does it take to write this number down using the place system?
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