what about the power property of natural logs… how is this derived?
ln(b^t) = t ln(b)
this is a bit confusing to me
@Luke: Thanks for the comments! I think I need to revisit the continuous growth explanation as it can be really confusing. I see continuous growth of r% meaning:
- 1 unit will product r% interest after 1 time period
- That r% interest, as soon as it’s made, will produce r% on itself after 1 time period
So, the “money that money earns, earns money”.
The problem is the difference between APR (annual percentage rate – your “growth rate”) and APY (annual percentage yield – the total amount you change as a result of the compounded interest).
Because it’s e^r*t, you could see it as 100% growth letting time vary (as you mentioned), or 1 year letting the growth rate vary. But still, I need to think about ways to clarify.
Thanks for the relation to chemistry, it’s been a while since I’ve looked at that and it’s nice to have reminders :).
@Anonymous: Great question. ln(b^t) means: How long does it take to grow to b^t?
Well, growing to b takes ln(b) time (for example, growing to 3 takes ln(3) time). Growing to b*b takes ln(b) + ln(b) time [grow to b, then grow to b again – to grow to 9, you grow by 3, then grow by 3 again].
So, growing to b^t means you just add up the time to grow to b [ln(b)] “t” number of times: t * ln(b). Hope this helps!
I’m taking a financial structuring course at an undergraduate level… always knew how to use e but never really grasped what the hell it meant! thanks a lot for this it really helped refresh and consolidate my knowledge of this function!
thanks again!
@RP: You’re welcome, I’m glad you enjoyed it! The true meaning of e didn’t hit me until much longer after I “learned” it also :).
Many research papers use the natural log for items such as total assets or sales revenue as a size variable. Why is the natural log a better size measure versus just using the actual dollar amounts for total assets?
Phil
Hi Phil, good question. I’d be surprised to see natural log taken on a straight number (total assets or sales) – usually it’d be used for a growth factor or a series. It could be used to rescale numbers to a “human” factor (like a Richter scale or Decibel lets us talk about large variations in a smaller range). But there may be another interpretation of the natural log which would make sense for the raw number.
U’m In Pre-Calc and sadly when the teacher talks thats all she does I get better explinations from other students, and this was a new topic, which of couse i thaught i would never grasp thank you for this…:(…i sorta wish you could teach me now…but i digress and return to theact of thanking you and hope that you cover some other topics i’ll have to come.
I just want to THANK YOU because your article was so just so much more straight forward then my text book. This article explained the concepts and a more comprehensive way then I think even my teacher could explain it. THANK YOU AGAIN and keep wrtiting!
@Kid Crew: Glad you enjoyed it! Hopefully I’ll cover the topics you’re learning about. Really happy to hear things are clicking.
@Calculous Crushed: You’re welcome! I had a similar reaction when learning about logs and exponents, and only “got” them much, much later after learning about them. Thanks again for the encouragement.
I dont quite get the the rule of 72 thing could explain that to me. I dont get the idea of having rate and time always getting the same answer.
Thank you. (great article, btw)
Can you please explain how log returns work (ie IBM starts the day at $5 and finishes the day at $6…the return would equal ln(6/5))…i believe this type of calculation is used in finance to calculate returns. This is different than (6-5)/5…I understand the log is using continous growth, but can you explain how to think about it, and also why most firms use one technique rather than the other.
Hi Jeff, interesting comment! I’m not certain of the reason, but here’s what comes to mind.
e and natural log let us discuss growth in a way that is “flexible”. Using your example, (6-5)/5 represents 20% growth in a day. ln(6/5) ~ 18%, which is the equivalent continuous growth rate (which is less than 20% because continuous growth earns “interest on interest” and can catch up).
With the continuous growth rate, we can say that having 18% growth for 1 day will have the same impact as 1% growth for 18 days. Or 2% for 9 days, or 3% for 6 days, etc.
If we use 20% standard growth, we can’t easily swap out the numbers since rate and time can’t be mixed. Natural log lets us convert rates to “standard” format which makes it easier to do mental “what-if” scenarios. That’s one application I can think of :).
[…] The time needed to grom from 1 to A is the time from 1 to 2, 2 to 3, 3 to 4… and so on, until you get to A. The first definition defines the natural log (ln) as shorthand for this “time to grow” computation. […]
Hi Kalid, brilliant explanation, really enjoyed it.
I was thinking about cell division.
If I understand correctly, we cannot use our formulas if there are less than 100 cell divisions over the time period under consideration; if there are less than 100 cell divisions the formulas will be too inaccurate.
If the cell could divide in zero seconds then our formulas using e would be 100% accurate.
A cell division every zero seconds corresponds to a growth rate of infinity%
Using our formula above The time for the cells to double would be given by
time x infinity = ln(2)
time = zero seconds.
This is a trivial result, but ironically this is the only time the formulas are 100% accurate.
The rest of the time, when we are dealing with real life systems, the formulas are so close to accurate that errors can be ignored.
Have I got it or am I way off the mark?
Yes yo, bn a great pleasure reading the article. It opened me up a bit, thanx to you proficiency.I wish to kindly ask if you can be able to explain to me how to hanndle exprassions of the form :Ln(x+y). Just how this can be simplified will do me good.
Thanx in advance!
@chris: Great question! Here’s the key:
Real-life systems can’t grow perfectly continuously. But, they can be approximated by some continuous function.
An analogy may be how we try to make wheels on a car “round” – that is, follow the equation of a circle (x2 + y2 = r2). Of course, at a molecular level the atoms aren’t exactly on that line, but the above equation describes the shape of the wheel very well.
For cell divisions, individual cells may not divide at a continuous rate (or even at 100 times per second). But the whole “lump” of cells may appear to grow at some rate – we have a lumpy goop that is growing over time.
e^x can then approximate the “expected” growth rate of that goop, like the circle can model the “expected” location of atoms in the wheel. So, you don’t need your cells to grow at any certain rate, you can find some rate that approximates the way it grows. You can find some growth rate x to make e^x fit the curve of your actual growth. If your goop grows 2x after a unit of time, your equivalent continuous rate is ln(2) = .69, and can be modeled by e^.69x. Hope this helps!
@Sehluko: Hi, I’m not sure there’s an easy way to simplify ln(a+b) by itself. If there are other terms ln(a+b) + ln(c+d) you can do = ln([a+b]*[c+d]).
Kalid,
Great work.
Question: as you know, we think of risk and return in finance. Let’s assume we have an expected rate of return (on which we can apply e assuming the rate is constant or close to constant). But let’s further assume that returns are uncertain and normally distributed around the mean (with a known variance). As my variance in returns increases, what happens to e?
Thanks!
David
Yay! I love math. Thanks to these wonderful lessons.
Thanks for your article. It helped a tremendous amount. My unit guide describe all of ln and e within one side of a standard a4 piece of paper. It was incredibly useful! My unit jumped from ln and e into exponential and logarithmic models (Ln Reg). Thank you very much, and I’d be happy to read more!