In general, you can flip the fraction and take the negative: $\ln(1/3) = – \ln(3) = -1.09$. 693 units (negative seconds, let's say) we’d have half our current amount. If we reverse it (i.e., take the negative time) we’d have half of our current value. Ok, how about a fractional value? How long to get 1/2 my current amount? Assuming you are growing continuously at 100%, we know that $\ln(2)$ is the amount of time to double. You’re already at 1x your current amount! It doesn’t take any time to grow from 1 to 1. What is $\ln(1)$? Intuitively, the question is: How long do I wait to get 1x my current amount? How’d they turn multiplication into addition? Division into subtraction? Let’s see. You’ve studied logs before, and they were strange beasts. With me? The natural log gives us the time needed to hit our desired growth. If we want growth of 20.08, we’d wait 3 units of time (again, assuming a 100% continuous growth rate). After 3 units of time, we end up with 20.08 times what we started with. $\ln(x)$ lets us plug in growth and get the time it would take.$e^x$ lets us plug in time and get growth.Now what does this inverse or opposite stuff mean? Speaking of fancy, the Latin name is logarithmus naturali, giving the abbreviation ln. The natural log is the inverse of $e^x$, a fancy term for opposite. $e^x$ is a scaling factor, showing us how much growth we’d get after $x$ units of time. For example: after 3 time periods I have $e^3$ = 20.08 times the amount of “stuff”.
![natural log matlab natural log matlab](https://sunglass.io/wp-content/uploads/2020/01/img41.png)
How much growth do I get after after x units of time (and 100% continuous growth).By converting to a rate of 100%, we only have to think about the time component: We can take any combination of rate and time (50% for 4 years) and convert the rate to 100% for convenience (giving us 100% for 2 years). As we saw last time, $e^x$ lets us merge rate and time: 3 years at 100% growth is the same as 1 year at 300% growth, when continuously compounded. Not too bad, right? While the mathematicians scramble to give you the long, technical explanation, let’s dive into the intuitive one. $\ln(x)$ (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.
![natural log matlab natural log matlab](https://ars.els-cdn.com/content/image/3-s2.0-B9781558608740500083-u07-46-9781558608740.jpg)
After understanding the exponential function, our next target is the natural logarithm.