Two Types of Growth

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:

Logarithmic Curve

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.


  • Warren Roberts

    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.

  • dag

    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.

  • Franklin Chen

    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.

  • Taylor M

    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.

  • Karen Renee

    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.

  • Mike

    Inspired by Antifragile perhaps?

  • Eddie Schodowski

    Martial arts is certainly logarithmic.

  • Scott Young

    Mike,

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

    -Scott

  • Harry @ GoalsOnTrack

    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.?

  • Bertie Steinein

    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.

  • Jeff Leenz

    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.

  • Michael

    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.

  • John Graves

    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

  • Mike

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

  • Tiny

    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/C….

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

  • Christian Kleineidam

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

  • Ryan

    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.

  • Evan

    @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.”

  • Ilham

    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/t

  • David

    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.

  • Franklin Chen

    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.)

  • Brad

    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?

  • Dan

    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.

  • Scott Young

    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

  • Eric

    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.

  • Miles Rosenberg

    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!

  • Lauren

    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.

  • Eng Wen

    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.

  • Scott Young

    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

  • Winifred

    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.

  • Vito

    Fascinating.

    As a person who wants to begin a new career in photography, i have noticed that my technical and artistic skills have tended to be logarithmic; i.e., learning composition, lighting, etc. have a fairly steep early learning curve. Initial progress was fast.

    The interesting part is, the business development part is much more exponential in nature (I hope!). That is, initial paying jobs are relatively few and far between, but I hope that once I am on the steep part of the curve I will be turning away opportunities!

  • David Hurst

    Congratulations on the mention in David Brooks’ column today! Here is the response I just posted on NYT:

    “Seeing yourself as part of a landscape…” these dynamics are fundamentally ecological and they apply to many human systems, especially commercial firms. The “S”-shaped sigmoid curve captures both the exponential and the logarithmic phases. In the beginning progress is slow and, as Benjamin Bloom’s work on talent development showed, one first has to develop the commitment necessary to work hard at developing virtuous habits. Once these disciplines are established, growth is exponential for a while (think Apple under Jobs II) until it slows, as a plateau is reached. Now it’s easy to fall into a competency trap and the organization may require a “shake-up” to break the constraints that bind it.. Nature uses wind and fire, flood and pestilence to sweep away decadent structures and create the conditions for renewal. This is the beginning of a reverse “S”-shaped curve. Add the two sigmoid curves together and you have a Moebius strip, a symbol of sustainability and renewal through a perpetual cycle of learning and selective forgetting, creation and destruction…

  • Bob Kocsis

    As a baby boomer and lifetime mechanical engineer, I see these simple growth (and learning) curves described by Mr. Young as useful in understanding my own 40 year professional career experience. This insight may help younger engineers or technical specialists as well.
    Like most engineers, I experienced exponential growth in science and engineering learning, and spent many hours mastering both calculus and mechanic’s tools, before I saw their intimate connection to my passions.
    For eight years after college graduation, my earnings experienced logarithmic growth, as my income exploded with each successful project, and then leveled off quickly. My career satisfaction has been logarithmic as well, but continuing in many wonderful ways over the years.
    Thank you all for the well written post, and the interesting comments.

  • Andrew

    Learning a language is logarithmic? Well I think it really depends on the language. If the language is closely related to your native language, there will be plenty of low hanging fruit to be found… similar sounds, similar grammar structures, words with shared roots, similar ways of clustering meanings into words (eg, blue=azul in Spanish, not blue or green=aoi as in Japanese).

    Try learning Japanese as a native English speaker and you’ll find that it’s a very exponential process. You get nowhere for a very long time… years can pass before TV news is intelligible. But once you have enough grounding in various areas they work synergistically. For example, knowing Kanji characters will give you a basis to more intelligently disambiguate the many homonyms you hear and also properly group them into words. Just a few similar sounds (kai, kei, kou, for example) may literally have dozens of possible meanings.

    English speakers tend to learn other European languages but it’s only once you move out of that language family do you realize how much you can’t take for granted. For example, if I can say “you look happy” in English, and duu means “to see or look” in Thai, can I necessarily say you “duu” happy in Thai? If it’s French or Italian you can make that kind of bet, but in Japanese it will mean “you observe (see) happy.” Things like that slow down progress in unrelated languages. (Thai, by the way, tends to group multiple meanings in fewer words so it is relatively quick to learn. If I’m not mistaken you “duu” happy works in Thai).

  • Scott Young

    Andrew,

    Agreed, I’ve found similar things in both Chinese and Korean. However, I’d point out that frequency distribution of vocabulary is not equal–there are many exceptionally low frequency words that all natives would still need to know in all languages. Because these volume of low-frequency words outpaces the high-frequency ones, and because pronunciation and prosody become harder to improve once they get “good enough”, languages must become logarithmic in the long run.

    -Scott

  • Jay Winograd

    Scott, I find your work fascinating, particularly as it pertains to my job as an instrumental music teacher. I have found that learning a wind instrument (trumpet or flute for example) is often fairly logarithmic in that progress is very rapid at first, but tends to slow down considerably after the first year of playing. Learning to play a string instrument, on the other hand, is much more exponential. Progress is often painfully slow initially, until it “clicks”. Assessing the learning styles of kids is crucial for determining which type of instrument will have more chance of success for any given individual. I have always advocated (with mixed results) that schools start string programs at a much earlier age. This way, when kids reach, say, seventh grade a trumpet player beginning at grade six will have roughly the same expertise as a violinist starting at grade three. Do you agree with this? Thank you for your feed back!

  • B. Anibal Palencia

    Is it accurate to say that things start out exponentially and then form a logarithmic curve? This is what my intuition tells me about this. I see this with Moore’s Law. The gains in computing are exponential, but as the supposedly dreadful end to Moore’s Law comes, we arrive at logarithmic growth.

  • B. Anibal Palencia

    Is it accurate to say that things start out exponentially and then form a logarithmic curve? This is what my intuition tells me about this. I see this with Moore’s Law. The gains in computing are exponential, but as the supposedly dreadful end to Moore’s Law comes, we arrive at logarithmic growth.

  • mantra cures

    I’ve been working on a Youtube channel for the past 18 months. The growth has been slow but exponential. At times, it is disheartening but I’ve managed to persevere and have a collection of 400 odd videos. Since the rate of growth differs from one month to the next and sometimes it is even negative, it is really hard to keep up the hard work. Should thank you for the exceptionally well-written article that boosts my hopes.

  • mantra cures

    I’ve been working on a Youtube channel for the past 18 months. The growth has been slow but exponential. At times, it is disheartening but I’ve managed to persevere and have a collection of 400 odd videos. Since the rate of growth differs from one month to the next and sometimes it is even negative, it is really hard to keep up the hard work. Should thank you for the exceptionally well-written article that boosts my hopes.

  • angelo vickers

    If exponential
    graph the graph curves up very quickly, what does it mean? If the graph curves up very slowly, what does it mean?

  • angelo vickers

    If exponential
    graph the graph curves up very quickly, what does it mean? If the graph curves up very slowly, what does it mean?

  • haidytarekzakaria

    My work has been-and I think still is- exponential. Starting a project that does not have an already established popularity in my country has been very challenging. 2 years in, we are just getting some traffic and traction to our project. I never had a name for the progress but I knew that it would be some time before I got to see any actual results. Thank you for shining a light on the difference between both curves.

  • haidytarekzakaria

    My work has been-and I think still is- exponential. Starting a project that does not have an already established popularity in my country has been very challenging. 2 years in, we are just getting some traffic and traction to our project. I never had a name for the progress but I knew that it would be some time before I got to see any actual results. Thank you for shining a light on the difference between both curves.

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