Comments on: How To Think With Exponents And Logarithms

By: Sreekanth

Sreekanth — Wed, 15 Apr 2020 06:40:17 +0000

Hi Kalid, i feel this is called real understanding(mapping the mathematical equations to realtime world). I really appreciate your passion and efforts. I have a question which may look silly but as many say no question is silly.
You mentioned
logarithm of change -> cause of growth.
Why u considered change = finalvalue/intial value(ln(14.4/9.9))
Rather we generally see change = final value- intial value ln(14.4 – 9.9) or
Change ratio = (final value- intial value)/intial value.
Eg: a change from 100 to 125 means
Change = 125-100=25
Change ratio = (125-100)/100= 1/4
Percentage change = 25%.
Please either correct me if am wrong or consider updating the aricle.
Thanks in advance.

By: Yash Arora

Yash Arora — Tue, 20 Nov 2018 05:26:00 +0000

In reply to kalid. awesome explanation !!!!!!!!!!!!!!!!!!!! ^_^ ^_^ ^_^

By: kalid

kalid — Sun, 21 Oct 2018 05:01:00 +0000

In reply to JOSH.

Hi Josh, great question. Exponents and natural logs mix rate and time in a way that can be confusing. Let’s start with a simpler scenario and work up.

Let’s say we doubled our money ($100 grows to $200). Without being specific about the rate or time yet, what can we work out?

Well, ln(200/100) = ln(2) = .693

This means a few things:

* If our continuous growth rate was 100%, this change would happen in .693 units of time. (If our rate was 100% per hour, we’d double in .693 hours. If 100% per year, we’d double in .693 years).

* Alternatively, if our growth rate was .693 units (69.3%), we would double in exactly one unit of time.

We can actually mix and match: total change = rate * time

If our time period increases by 10x, our growth rate divides by 10x. (So 1 year of 69.3% growth = 2 years of 34.65% growth = 10 years of 6.93% growth)

Phew. So the natural log gives a quantity which represents “rate * time” and often, we pick one to be 100%. If we assume rate is 100%, then the natural log = 100% * time. (e^x = e^{100% * x}, and assumes a 100% growth rate by default).

So in summary, ln(14.4/9.9) = .374 is the total “rate * time” that happened. We do:

.374 = rate * time

and plug in time=10 years to get

.374 / 10 years = rate

By: JOSH

JOSH — Sat, 20 Oct 2018 14:40:00 +0000

In your other articles about natural logs, you explain that taking the natural log of a growth rate, gets you the time it takes to achieve this growth. In your example above about economic growth, you take the natural log of the percent change (14.4/9.9), or growth of the economy between 2000 and 2010, and say this returns a rate. However, following your description on the article about natural logs, I expected this calculation to return the time it would take to achieve this growth.
How can we intuitively understand what this ln(14.4/9.9)=0.374 is? You say that 0.374 is the “cause of growth rate” in this example, however, you said that the natural log of a rate is supposed to be the time it takes to get to that rate. This is further assumed to be a rate when you divide by 10 years to get the “rate of growth” per year. I am confused because in my mind, this would have been taking time, returned by the natural log calculation, (0.374) and dividing by another time (10 years).
Please help me understand where my gap in logic is. Thanks!