I need some help understanding how logarithms work. I just can’t wrap my head around the concept, and I can only find videos online explaining how to rearrange a log equation. I’m looking for an explanation as to how it works, and perhaps the theory behind it.
I’ve also heard that log is easier to use for more complex calculations. How does rounding work with log? What is a natural log? What is e?
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While everyone already said, what the log is, I’ll tell you **why** we need it.
The most important property of logarithm is that it replaces multiplication with addition: `log(a*b) = log(a) + log(b)`. In other words, it replaces relatively difficult operation (`*`) with a simpler one (`+`).
Of course, it works with related operations as well:
* inverse multiplication (division) gets converted to inverse addition (subtraction): `log(a/b) = log(a) – log(b)`
* repeated multiplication (power) gets converted to repeated addition (multiplication): `log(`a^(b)`) = b * log(a)`
* n-th root gets converted into division by n.
That’s how people did calculations from 1500s to computers: they’ve used log tables and log slide rules.
Natural log is “log base `e`”, but most importantly – it’s “nice calculus log”. The derivative of natural log is just `1/x` (derivative of other logs is `(some_number)/x`). In fact, this is actually an ELI5 definition of `e`: “`e` is such a number, for which e^(x) and log_e(x) have nice and easy derivatives”. It just so happens, that such number is `2.718…`.
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