Inverting Exponentiation? Why Logarithms and Roots Are a Bit Weird.
What's the difference between logarithms and roots, and what's more, why does there have to be a difference at all?
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It's natural to presume that exponentiation should be "undo-able" in the same way that operations like addition and multiplication are, namely, with a single inverse that returns either of the original addends or factors. However, when we try to apply the inverses of exponentiation in this way, we run into problems. For example, if we don't know the base of an exponential equation like x^3=125, we can recover that base by taking the cube root of each side of the equation. But if instead it were the exponent that was missing, a root would not help us. In an equation like 5^y=125, we can't simply take the "y-th" root of each side, because it's precisely y that we do not know. Instead we need a different sort of inverse called a logarithm. Ultimately, the reason that exponentiation requires two different sorts of inverses, when operations like addition and multiplication do not, is that while addition and multiplication are commutative, exponentiation is not.
And my apologies to @3blue1brown for messing up his channel's name. I don't know what I was thinking!
#ExponentiationBasics #LogarithmsExplained #RootsVsLogs
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