![]() $\ln(x)$ (Natural Logarithm) is the time to reach amount $x$, assuming we grew continuously from 1.0. ![]() $e^x$ is the amount we have after starting at 1.0 and growing continuously for $x$ units of time.Don’t see why it only takes a few years to get 10x growth? Don’t see why the pattern is not 1, 2, 4, 8? Read more about e. If you want 10x growth, assuming continuous compounding, you’d wait only $\ln(10)$ or 2.302 years. Suppose you have an investment in gummy bears (who doesn’t?) with an interest rate of 100% per year, growing continuously. Given how the natural log is described in math books, there’s little “natural” about it: it’s defined as the inverse of $e^x$, a strange enough exponent already.īut there’s a fresh, intuitive explanation: The natural log gives you the time needed to reach a certain level of growth. After understanding the exponential function, our next target is the natural logarithm.
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