What is it about the number e that makes it so that taking log to the base e is called the natural logarithm?
In: Mathematics
We use base-10 logs sometimes because we have 10 fingers and count in tens. log base-10 tells us the magnitude of a number in powers of how many fingers we have. Any other log base would be equally arbitrary. What would be a “natural” base, one free of human biases?
Well, mathematicians decided “e” would do that nicely. e can be calculated in lots of mathematical ways that don’t depend on ten fingered creatures and decimal arithmetic, for example as the infinite sum of inverse factorials. “e” appears “naturally” in a lot of mathematical areas, which we’d expect seven-fingered aliens who count in base-14 to also find. For example the rate of change of e^x (e to the power of x) is e^x, and the rate of change of the natural log function is 1/x, which looks simple and a “natural” choice for log base. Other bases end up with constants and scale factors when you try and compute the rates of change (slope, gradient, differential).
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