How To Think With Exponents And Logarithms

This is a companion discussion topic for the original entry at

@Ashok: Thanks! I have an intro to linear algebra & matrices here:

@Ben: Thanks, I’d like to do some collaboration in the future too.

@Richard: Great question, happy to clarify. You may want to check out this article:

which goes into the scenario in more detail.

The “grower’s perspective” is the growth rate experienced by a single component, whereas the “observer’s perspective” is the growth rate after all compounding effects are taken into account.

Let’s say I invest $1000 at 2% a month.

An individual dollar just knows to grow 2% each month. At the end of the year, it thinks “I have contributed 2*12 = 24 cents to the whole”. That is true for direct contributions, but its interest earns interest! And that earns interest, and so on :).

So, to an outside observer, the total effect of “2% return every month” is really:

(1 + 2%)^12 = (1.02)^12 = 1.268 = 26.8% annual return

There are two “rates” we can describe in this scenario: the 24% rate each dollar intends on contributing over the year, or the 26.8% growth that we see after all compounding effects are worked out.

Let’s say we grew for 5 years. We could describe this as both:

Grower’s perspective: (1 + 2%)^(12 * 5) [“Take the individual monthly growth and apply it for 5 years.”]

Observer’s perspective: (1 + 26.8%)^5 [seen as “Just take the final annual growth and apply it for 5 years”]

You can hop between one perspective and the other, in science it’s usually easier to think about the contributions on a per-item basis (and then work out the result), and in finance we may just care about the final result (and don’t care about the details for what each dollar plans on contributing).

@Michael: Awesome, glad it helped.

MUCH more helpful in explaining the computation of growth rates in finance.

Thanks Kalid!

Hey Kalid- I don’t know if you recall me, but I’m moving back to Seattle!

I love this article- I was wondering what your intuition was on logistic curves, logistic regressions?

I often think of them as “depressed exponentials”- stuff that gets increasingly bogged down with whatever is limiting their growth. I think I’ll be studying more of this class of equations in my new job, but wanted to get your thoughts.

Hey Zach, that’s great! I’m thinking of doing a Seattle-area intuitive math meetup and I’ll put you down :).

You know, I haven’t really studied logistic curves very much, but I like that description. From looking around, here’s a quick stab:

A regular exponential has the property

d/dx f(x) = f(x)

that is, we’re always growing by 100% of the amount that we currently have. Resources in the environment seem unlimited – however much we have, there is enough for us to crank out helpers, who make helpers, etc.

If our resources get constrained as we get further along, it seems we’d describe our growth like this:

d/dx f(x) = f(x) * percent of growth factor remaining

Since we’re dealing with percentages now, f(x) isn’t a function of raw numbers, but a percentage of the total you can possibly have. As you reach for 10% of your possible max, you are growing by 90% of your current amount. As you get to 25% of your possible max, you are growing at 75% of your current amount.

From here it seems like we get the logistic curve:

d/dx f(x) = f(x) * (1 - f(x))

For me it seems the main shift is that f(x) is now returning a percentage whereas with regular e^x, we assume we’re getting an absolute amount (there is no max we’ll ever reach).

Thanks for your article. I don’t know if it’s because english is not my native tounge or I just don’t get it. Can elaborate a bit more on the grower vs the oberver viewpoint. I’m having trouble understanding it.

Brilliant, Kalid. Is this your day job?

I understand e, as a rate of change over time, taking account of the fact that the quantity which has grown, itself grows…very nicely explained by you. What does In mean? as for example In(x) is the time to grow to x. How should I read In…is it integrate?

@Dan: Thanks, glad you enjoyed it! My day job is actually doing web development, but I’m hoping to make this site an ever-increasing part of my life :).

@Jonathan: Whoops, I should have clarified that. ln is short for “logarithmus naturalis”, the fancy Latin name for the Natural Logarithm. I’ll fix up the article with a note, thanks for the feedback.

We “studied” logs in a trig course in 1949 at Columbia U. Back then “we” were burdened with log tables. Not a single student in the class had any idea of what was going on or what a “characteristic” or a “mantissa” was. The nearest we could make out was that it was like saying “Shazam” produced Captain Marvel, i.e., nothing made sense but if you do the right things you pass the course. (Also like “Open Sesame!” yelled at the rock in front of the cave.) But, as the old folks say, “Better late than never.” Thank you for your extraordinary hard work and insights. I also love the design of your site. You are indeed an ace web site designer.

Hi George, thanks for the note! It seems not much has changed, when I was in school we also learned things by rote. It’s now a decade later, and I’m finally making sense of the ideas I learned.

I’ve started to realize learning was never on a strict timeline, you never have "just 4 years to internalize a concept, and there’s always new intuitions to uncover when we dust off an old idea. (Also, glad you’re enjoying the site design!)

I hope this site gets increasing successful. Its value is innumerable. :slight_smile:

Thanks Zenko, I appreciate it :).

You are fantastic , you explained exponential and logarthmic beautifully, i am waiting for matrices Kalid

Have to talked to the Khan Academy people. I would love to see Vi Hart and you do some videos together.

There is no time I surf that I will not check I graduated from one of the universities in Nigeria in 2003, I did not have the confidence to go for postgraduate until I buried my head 3years from 2009 and internalised the once recondite maths ideas, before going for postgraduate study in electrical engineering. This article is again another wow!!!. Thanks for the global inspiration that you give to all the scientists in the world.

Thank you so much for these illuminating articles! It makes thinking about exponents and logarithms natural.

There is one thing that seems odd to me though. It is the use of ‘cause’. It seems that ‘rate’ is what is meant. Is it from a metaphor you were using somewhere else?

Hi Tom, thanks for the note. I see cause as both the rate, and the time we allowed it to grow [sometimes this is implicitly one period of time].

For example, we observed a bacteria colony that started with 1,000 members and grew to 4,000 over the course of 3.5 days.

The “cause of this change” was that some growth rate was applied for some amount of time (39.6% continuous growth, applied for 3.5 days, in this case). The effect was that 1,000 members grew to 4,000.

We can say cause = rate, but only if we’re certain we’re dealing with one unit of time.

Thanks for this but topic is far and away above my basic level. I haven’t had any reason to use logarithms in the past 50 years and have never used natural logs. Keeping on trucking. :wink:

It made me think differently about exponents and logarithms . I gained lot of insight about e and natural logarithms