Anything you try to improve will have a growth curve. Imagine you ran everyday and you tracked your speed to finish a 5-mile course. Smoothing out the noise, over enough time you’d probably get a graph like this:

Here, improvement works on a logarithmic scale. As you get better, it gets harder and harder to improve. Elite athletes expend enormous effort to shave seconds off their best times. Novice athletes can shave minutes with just a little practice.

Logarithmic growth is the first type of growth. This is where you see a lot of progress in the beginning, but continuing progress is more difficult.

Now imagine a different graph. This time you’ve build a new website you update regularly and you’re measuring subscribers. This graph would likely look very different:

This is exponential growth, the second type of growth. Website traffic is often exponential because as a blog attracts more readers, there are more opportunities for word about the blog to spread. A blog with zero traffic also has zero word of mouth.

I’ve noticed most things tend to be either logarithmic or exponential growth. Despite this, linear progress is what most people expect. We tend to expect things to move in the same direction or rate as they have in the past. This violation of our expectation leads to some mistakes in how we set goals and act on them.

**The Logarithmic Growth Mistake**

The first kind of mistake is assuming straight-line growth, when reality is actually logarithmic. There are many situations which usually fit this pattern:

- Athletic performance
- Weight gain/loss
- Learning a new language
- Productivity
- Mastery of a complex skill

Assuming straight-line growth means overconfidence in long-term progress. As a result, it is easy to hit plateaus if the difficulty isn’t deliberately tuned to break your comfortable rhythm.

Logarithmic growth also implies it is easier to slide back down the hill. Since it is so steep in the beginning, carelessness can mean those immediate gains are often easily lost. Losing weight quickly may be more desirable than losing it slowly, but it also risks putting it back on again quickly if you stop your efforts.

Logarithmic mistakes are common, but so to are mistakes when reality is in the other type of growth curve.

**The Exponential Growth Mistake**

Once again, people view progress linearly when it is, in fact, exponential. Some examples which usually follow exponential curves for at least part of their lifecycle are:

- Technological improvement (e.g. Moore’s Law)
- Business growth
- Wealth
- Rewards to talent/career

Unlike logarithmic curves, almost nothing is consistently exponential. Most are only exponential over some range of values, outside of which they are logarithmic again.

No business reaches near-infinite values, even though this would be implied by an exponential curve. Eventually market share is saturated or competition stabilizes growth. However, for many types of businesses, exponential growth can persist for much of the business lifecycle.

Exponential curves are somewhat rarer than logarithmic ones, however the mistakes here can be even more costly. Expecting linear growth when it is actually exponential causes many people to give up way before they should.

Several years ago, I remember being disheartened when drawing straight-line projections of my business income. At the going rates, it was often a dozen years away or more before I could make a full-time living at it. However, my income turned out to be exponential. Despite spending more time below my ideal projections, I ended up eventually surpassing them.

Exponential areas of life are full of quitters. People making linear assessments of viability and giving up before the exponential curve can take hold. Not all of this is irrational, many exponential areas are high-variance as well. However, the problem with exponential domains is that the feedback can often look bad, even when it is good.

**Is Your Growth Exponential or Logarithmic?**

Both types of mistakes, failing to recognize exponential domains and logarithmic ones, are costly. It’s not always clear which is which, but by reasoning about the features of a domain you can get a better sense which one it is likely to be.

The easiest way to tell is to look at how other people have progressed in that field. Don’t pay attention to their rates, just pay attention to the shape of their growth trajectory. Is it the kind that slows down with mastery or speeds up?

Exponential growth stories tend to be the ones where a person struggled long and hard with little to show for it, then started quickly gaining success. These stories often seem to be overnight successes, since they ignore the years of obscure toiling.

Logarithmic growth stories involve a continuous dedication to remaining at your peak. A fluent speaker of a second language will comment on the regular practice required, not the sudden moment where it all “happened”.

Other features of the environment can tell you whether something is exponential or logarithmic. Exponential environments often seem to be based on a catch-22 or circular causality. Having money makes it much easier to make more money, not only because of interest rates but because people give rich people opportunities they wouldn’t to those without success.

Logarithmic domains usually have diminishing returns once the “obvious” solutions are taken. The more you work at them, the harder you have to look for insight to generate new gains. Occasionally you can discover overlooked opportunities and regain the steep part of the curve, but this is inherently difficult.

**Growth Mindsets**

In logarithmic domains, two mindsets are important. In the beginning, high-growth phase, the emphasis needs to be on maintaining long-term habits. Since growth is fast initially, care needs to be taken so that it won’t slide back down once effort is removed.

In the later, low-growth phase, the emphasis needs to be on habit breaking. Since low-growth is often caused by calcifying routines, deliberate effort needs to be taken to break out of that comfort zone.

In exponential domains, the mindset of resilience and endurance are critical. Since feedback is sparse and generally negative during the initial part of the curve, it takes dedication to persist. Part of the reason, entrepreneurs are often consumed by their own vision is that it helps block out the negative feedback until they can reach the exponential part of their growth.

Now it’s your turn. What are you working on right now? Is it more exponential or logarithmic? What features of your environment do you believe make it that way? Share your thoughts in the comments.

My favorite post! Simple yet profound.

I recently started self tracking a number of things in my life. One of them was the effect of 10 minutes of intense exercise and 800mg of DHA of Omega 3 on brain function (separate tests of course).

The results were interesting but it was the side effect that really got me excited. I went through a very clear logarithmic curve to reach a baseline of the arithmetic test I used to measure the results, before I started the ABA experiment.

I don’t know whether most things are logarithmic or not, but it feels like that. I think motivation has a leading role in that, since if it’s sparse, you won’t improve easily, no matter you’re at the very beginning.

I am currently struggling to learn German. It should be a fairly logarithmic curve, and I think it is but… oh my, it turns into a beeline every time I look aside. Conversely, my English learning curve is undoubtedly logarithmic.

Anyway, I’m most interested in your opinion about ‘quitting too soon’ when you are at an exponential curve. I hope you develop that argument further.

An interesting contrast, but here are some additional ideas. In long-term deliberate growth, I think that plateaus are natural (because performance measures do not capture inner refactoring that may take a while to actually manifest itself), and more important, apparently backsliding is also natural (as one unlearns something in order to learn something more correctly). Often there will be little “exponential” bursts when something suddenly clicks, and then a plateau or maybe logarithmic growth.

I would argue many things follow a typical cell growth curve. Many cell cultures have a lag phase where it doesn’t look like a lot is happening. Then they begin to enter the exponential phase. They continue on the exponential until something changes and they begin to enter the stationary phase.

You hinted at this when you mentioned no business or skill can go to infinity. Something is going to be limiting. In cell culture, it’s usually nutrient deprivation or some sort of contact inhibition. But in business it could be market saturation, loss of mission focus, or any of the myriad things that could stifle the growth of a business.

It’s definitely good to figure out what kind of growth curve you’re on so expectations match with reality. Thanks for the great article, Scott.

You’re right. Developing art skills is logarithmic. There are many possible approaches that weave together to create a stronger structure for further improvement. If I ever reach a peak, there’s more to learn elsewhere. This attitude is the prerequisite for having a growth chart at all, I suppose.

But, selling my art is more likely a matter of exponential growth. The amount of people who value it increases with exposure,… but I’ll always be creating new work and growing, so my style will change.

Hmm … you lose characteristic elements of your work when you grow, don’t you? I’m trying to enjoy where I’m at as much as possible along the way, since I have already passed beyond the style of the drawings from when I started.

Inspired by Antifragile perhaps?

Martial arts is certainly logarithmic.

Mike,

I actually haven’t read that book yet, but I should!

-Scott

Quite interesting. For exponential growth, how does it explain some product or business suddenly catches on, even in the early stages, such as facebook, twitter, etc.?

Your first graph doesn’t look right. Time to run a 5 mile course would decrease as you got better, not increase. So what you want to plot is speed.

I don’t know if it’s allowed to suggest online courses here, but I think it is a must see for those who is interessed in growth models in general.

The course is ‘Model Thinking’ from http://www.coursera.org . As you have completed the course, you’ll be able to reason over simple models such as how a gossip spreads through more complex models such as why chinas is stop growing. You don’t have to pay any fee do take the course. It’s tottally free.

I love this article. It explains why I have hit a plateau in my language learning. After learning the basics I have noticed that I am having to put more time in to learn the small details that are not essential but just a nuance.

This reminds me a lot of the chart Tim Ferriss used in 4HourChef. It’s good to know there is a specific point where you become an “expert” before your ROI goes to crap and then just maintain the skill as it is.

These ideas do look very much like the convexity discussed in Nassim Nicholas Taleb’s book Antifragile. Also worth a look: The Success Equation: Untangling Skill and Luck in Business, Sports, and Investing by Michael J. Mauboussin

I’ve always thought most things followed the sigmoid curve. Fast start, then things slow down.

One more vote for supporting Mike, I was thinking of the same. I just remember it as cumulative distribution function (http://en.wikipedia.org/wiki/Cumulative_distribution_function).

Exponential or logarithmic growth seems to me just part of the whole curve.

[...] Two Types of Growth—A great, great post by Scott H Young on how we should approach growth. [...]

In biology most growth is sigmoid: (http://en.wikipedia.org/wiki/Sigmoid_function). A sigmoid function can look like it’s exponential at the beginning and logaritmic at the end but it’s not.

This is so timely.

I’ve recently hit a bit of a plateau in my learning of a new subject. When I first started it was all beginner stuff and as such I picked it up pretty quickly. Now however I’ve moved into more intermediate/advanced concepts the amount I’m learning is considerably less.

Has made me re-evaluate how much further I want to go, knowing that to reach the next levels are going to take a lot more work to get the same level of achievement as in the beginning, especially given that this isn’t a topic I particularly want to master, I just wanted a better understanding of the basics and to be able to understand a bit better the more advanced.

Seems obvious now I type that out, but this post brought it home to me.

Many thanks Scott.

@Mike @Christian

He described the sigmoid-like nature of ‘exponential growth’ in real world situations…”Unlike logarithmic curves, almost nothing is consistently exponential. Most are only exponential over some range of values, outside of which they are logarithmic again.”

Hello Scott,

Great article. I am writing this as a response to Franklin Chen’s comment, but also something you might like to check out if you haven’t already.

It’s a book entitled: “Mastery” by George Leonard. He describes 4 types of people: Dabblers, Obsessives, Hackers, and Masters. What’s particularly interesting is that he uses the sigmoid curve as a framework for his theory. Particularly that most people who attain mastery are stuck most of the time on plateaus (logarithmic) but that through hard work and “internal refactoring” (Franklin) they hit exponential curves, quickly reaching the next plateau of their journey. Starting the whole process again.

He also describes these people as loving the journey more than the results.

Here is a good link with some excerpts from the book:

http://www.mydrivingseat.com/the-blog/dabbler-obsessive-hacker-or-master/

If growth can be reasonably modelled as a logarithmic curve then knowing that helps you to make wise choices about what level of mastery you want.

I learned to juggle three balls, and acquired a repertoire of around 20 3 ball tricks. This progress doesn’t require any practice to maintain – I can quickly regain the same level of mastery after a break of several years.

However, to progress to 4 balls required an exponential amount of effort and then regular practice to maintain the skill. I wasn’t interested in that, so I stopped having taken that skill as far as I wanted.

Also, I wonder if character development has exponential growth? If I become more loving, present, and curious – initially this is difficult and makes little impact on my life, but there comes a point where these character traits open up more and more doors and create new, interesting connections.

My main takeaway from this article is to persevere in developing my character.

George Leonard’s book “Mastery” is very good, definitely recommended! (Taleb’s “Antifragile” is fascinating, but I still haven’t finished slogging through it yet.)

I need help with remembering mathematics formulas, case study detais and all the regulations. I am studying Occupational Health and Safety and I am find it difficult to remembering everything.

Can anyone offer advice on these specific areas?

Very interesting post.

I experienced this in poker, where the payoff is somewhat exponential when moving up stakes (it’s just like investing money with an edge).

However, it can take a very very long time to actually make a profit in the first place and break even. Only then the exponential growth starts. Most players give up before they reach this point.

[...] we have Scott Young who reminds us that not all growth is the same. In general, there are actually two types of growth. There is logarithmic growth where you get a lot of progress in the beginning, but it becomes [...]

Bertie,

You’re right–I changed the wording of the introduction to match the graph.

Christian,

Sigmoid-growth has both a convex and a concave phase, so as such it behaves somewhat exponentially and then somewhat logarithmically. Considering this article is meant as a mental rule-of-thumb, I felt such a simplification made the argument more clear.

There are also many different growth curves–polynomial, factorial, sub-exponential, sigmoid, logistic, etc. but the article centers on the rather unintuitive consequences of a non-linear growth which is the easiest to mentally compute.

-Scott

This is a pretty neat way of looking at how we develop skills. Can’t say it’s something I’m an expert on, but it’s intuitive and you explain the whole exponential/logarithmic growth rates and their differences pretty well.

Hi Scott, I really like this post. I also want to piggy-back on what Franklin Chen said about how some backsliding is natural, to unlearn and re-learn as we go. That said, having full commitment to change is still important. I learned this both from Charles Duhigg’s book and one that I wonder if you have read, Scott, called Making Habits, Breaking Habits, by Jeremy Dean (who keeps a site called Psyblog). One technique described in Dean’s book is chaining a positive habit to an earlier positive habit. If you do them sequentially, there seems to be a multiplier effect. Just wanted to share. Keep it up, Scott!

I am currently participating in a course that will, at its conclusion, bring the participants into a higher cognitive level. One of the papers we were provided to read is Adult Cognitive Development: Dynamics in the Developmental Web. It mentions the point that we slide back and then progress when we learn anything new and that it is a natural part of the process. It’s like the anecdote goes that when someone graduates from Harvard Business School and gets that first job, he is told to “forget everything you’ve learned in school about business.” It’s whole new world and whatever they’ve learned isn’t enough to get to the stage of actually living that reality. It also mentions that progress isn’t linear like climbing a ladder, but rather like a web. Thanks for writing about this topic; it helps strengthen my understanding.

I use Anki for language learning, and the cards for a given language EXPLODE to a large number when getting from beginner fluency to intermediate.

In the long-term, to reach native fluency, is it a good idea to BREAK doing Anki flashcard reps? What are the long-term strategies? Lots of Anki-ers don’t seem to answer that.

Eng Wen,

I tend to agree. Anki seems to be a good tool in the early intermediate level, where you can make basic sentences, but need vocabulary to fill them.

-Scott

[...] Young wrote a good post about two types of progress when learning something new: logarithmic and [...]

[...] activities have an obvious bootstrapping component. This is often why they experience exponential growth over some range of their progress—the effects create the causes resulting in a compounding [...]

[...] Favourite post: Two Types of Growth [...]

[...] reason might simply be that you’re at an elite performance level in a domain that experiences logarithmic (or some form of sublinear) growth. Elite athletes have low growth ratios precisely because their [...]

[…] advancement and subsequent disruption of media, and all sorts of other industries, are on an exponential growth […]

This consolidates what i learnt recently when i shared about how i felt at my current job. It feels like i have reached such a level that the environment does not allow me to grow any further (logarithmic). I was contemplating starting my own business and indeed the same view of exponential curve was explained by my mentor who said that it is surely going to be hard and slow in the beginning but with your own business the plateau that result in formal employment may not occur; as depending on how committed you are with your business its bound to be exponential growth.