Hi. Hope you liked my last 'intuitive' take on limits. This time iam sharing an awesome intuitive explanation of natural log. I strongly recommend having a look at the pdf( unedited version with diagrams) named 'Intuitive logarithms' ( I couldn't do the justice here ). Also explanation for exponential is available in pdf to go through before going through logarithms.( It has some graphics added in it for better understanding which i can't reproduce efficiently here). So lets start- DEMYSTIFYING THE NATURAL LOGARITHM (ln) Intuitive explanation: The natural log gives you the time needed to reach a certain level of growth. Suppose you have an investment in a bank (who doesn’t?) with an interest rate of 100% per year, growing continuously. If you want 10x growth, assuming continuous compounding, you’d wait only ln(10) or 2.302 years. E and the Natural Log are twins: e^x is the amount of continuous growth after a certain amount of time. Natural Log (ln) is the amount of time needed to reach a certain level of continuous growth. E IS ABOUT GROWTH The number e is about continuous growth. 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. Intuitively, e^x means: How much growth do I get after after x units of time (and 100% continuous growth). For example: after 3 time periods I have e^3 = 20.08 times the amount of “stuff”. e^x is a scaling factor, showing us how much growth we’d get after x units of time. NATURAL LOG IS ABOUT TIME: The natural log is the inverse of e, a fancy term for opposite. Speaking of fancy, the Latin name is logarithmus naturali, giving the abbreviation ln. Now what does this inverse or opposite stuff mean? e^x lets us plug in time and get growth. ln(x) lets us plug in growth and get the time it would take. For example: e^3 is 20.08. After 3 units of time, we end up with 20.08 times what we started with. ln(20.08) is about 3. If we want growth of 20.08, we’d wait 3 units of time (again, assuming a 100% continuous growth rate). With me? The natural log gives us the time needed to hit our desired growth. LOGARITHMIC ARITHMETIC IS NOT NORMAL: You’ve studied logs before, and they were strange beasts. How’d they turn multiplication into addition? Division into subtraction? Let’s see. What is ln(1)? Intuitively, the question is: How long do I wait to get 1x my current amount? Zero. Zip. Nada. You’re already at 1x your current amount! It doesn’t take any time to grow from 1 to 1. ln(1) = 0 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. If we reverse it (i.e., take the negative time) we’d have half of our current value. ln(.5) = – ln(2) = -.693 Makes sense, right? If we go backwards (negative time) .693 seconds we’d have half our current amount. In general, you can flip the fraction and take the negative: ln(1/3) = – ln(3) = -1.09. This means if we go back 1.09 units of time, we’d have a third of what we have now. Ok, how about the natural log of a negative number? How much time does it take to “grow” your bacteria colony from 1 to -3? It’s impossible! You can’t have a “negative” amount of bacteria, can you? At most (er… least) you can have zero, but there’s no way to have a negative amount of the little critters. Negative bacteria just doesn’t make sense. ln(negative number) = undefined Undefined just means “there is no amount of time you can wait” to get a negative amount. LOGARITHMIC MULTIPLICATION IS MIGHTY FUN: How long does it take to grow 4x your current amount? Sure, we could just use ln(4). But that’s too easy, let’s be different. We can consider 4x growth as doubling (taking ln(2) units of time) and then doubling again (taking another ln(2) units of time): Time to grow 4x = ln(4) = Time to double and double again = ln(2) + ln(2) Interesting. Any growth number, like 20, can be considered 2x growth followed by 10x growth. Or 4x growth followed by 5x growth. Or 3x growth followed by 6.666x growth. See the pattern? ln(a*b) = ln(a) + ln(b) The log of a times b = log(a) + log(b). This relationship makes sense when you think in terms of time to grow. If we want to grow 30x, we can wait ln(30) all at once, or simply wait ln(3), to triple, then wait ln(10), to grow 10x again. The net effect is the same, so the net time should be the same too (and it is). HOW ABOUT DIVISION?: ln(5/3) means: How long does it take to grow 5 times and then take 1/3 of that? Well, growing 5 times is ln(5). Growing 1/3 is -ln(3) units of time. So ln(5/3) = ln(5) – ln(3) Which says: Grow 5 times and “go back in time” until you have a third of that amount, so you’re left with 5/3 growth. In general we have ln(a/b) = ln(a) – ln(b) I hope the strange math of logarithms is starting to make sense: multiplication of growth becomes addition of time, division of growth becomes subtraction of time. Don’t memorize the rules, understand them. My gallery link: https://www.urbanpro.com/delhi/pankaj-k/2531974 Thanks for your time. Regards.

Asked by Pankaj Kumar 27/05/2015 Last Modified   06/10/2025

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