Information Science: Exponential, Logarithmic, and Linear Curves

Motivation

I often hear people use the word "exponential" in phrases such as "exponential growth" or "x is exponentially more difficult than y" (which often times doesn't actually make sense) but I rarely hear people use the words linear or logarithmic.  If you value using English correctly, they're all helpful to know and all are simple to learn.  See the Final Note below for more.

The Curves

For use in conversation, it's not necessary to know the math behind these curves, so I won't go into it.  Here are the curves, with exponential in blue, linear in red, and logarithmic in green:*

Pure math is hard to explain without examples.  I didn't grok algebra until I took calculus.  I didn't grok calculus until I took physics.**  I'll give a couple examples to explain these curves.

Learning:

Learning curves are often described by these curves.  I think this was the impetus for this blog post.  Note: This is kind of sciencey, that is to say, it's pretty nonscientific.  That's also to say I bet some people will disagree.

Exponential:

"Easy to do, hard to get very good at"

Example:
Shooting pool.  Pretty much anyone can pick up a pool cue and knock the balls around.  It might take you a half hour to finish a game of eight-ball, but you can do it.  In my experience, to get to even a "hey, you're pretty good" level takes dedication.  It's an incredibly nuanced game and just seeing the right shot to take, let alone making it and leaving the ball set up for the next shot... it ain't easy.

Logarithmic:
"Hard to pick up, easy to get pretty good at"

Example:
Water skiing.  It's not the most relatable thing.  If you've done it, you know that the hardest part is getting your butt out of the water to a standing position for more than 3 seconds.  I've seen people spend hours and hours getting pulled up, then flopping.  Get pulled up, wipe out.  Get pulled up, eat it.  It's pretty hilarious.  And by "I've seen people" I mean "I've been people."  But once you can consistently get out of the water, skiing itself, going outside the wake, that stuff isn't all that tough.  For this example, let's ignore how hard it is to get to the level of a pro.

Linear:

The more time you put into it, you just keep getting better.

Example:

Boy, this is a hard one.  I think whatever I say, people will challenge, so I'll say playing guitar.  It'll take about an hour for someone to learn the chords G, C, and D and they'll be able play dozens of songs, many of which are songs everyone knows.  Developing your barre chords takes some work.  Learning scales takes more.  Being able to improvise a solo takes more.  Being able to solo in D flat mixolydian in 7:4 takes even more.  There's always another hill to climb, but with dedication, you can do it.

Scaling:

In many systems, scaling up can be pose some of the biggest challenges you'll face.  Let's take a few real-world examples of scaling where these curves might apply.

Linear:

The more you add, it just keeps getting more difficult about how you'd expect.

Example: 

Chopping wood.  

If you have 100 logs to chop, maybe it'll take you an hour.  If you double the number of logs, you double the time.

Exponential:
You add a little it might be ok, you add more it's difficult, you add even more it's nearly impossible.

Example:
Making a reservation at a restaurant.
You have 4 people, not a big deal.  You have 10 people, ok that'll take some time - you should call ahead and they might have to rearrange the tables when you get there.  You have 30 people, you should definitely call ahead, some places might make you pay extra for a private room, and some places will tell you not to come.

Logarithmic:
You add some and it gets difficult pretty fast, but then you keep adding more and it's really not much harder.  Or, at first it's hard but as you add more, even a lot more, it's not much harder.

Example:
I had a hard time coming up with an example for this.  I guess the most obvious would be something that takes a lot of setup, but then isn't that hard to actually do, or after the setup, the work gets easier.  This example doesn't feel as good as the others, but I'll go with automation.

The linear example was chopping wood, and when you double the wood, you double the work.  If you decided to build a wood chopper, getting that very first log chopped would take a huge amount of work because you have to design and build the machine for that first long.  But once the machine is operational, just feeding some logs into the hopper just takes a little bit of work.

Final Note:

I appreciate correct use of English.  That's why I get frustrated when people say something like "the Titanic is exponentially bigger than that sailboat over ther."  Actually, it may be linearly bigger, it depends on the data points between those 2 objects.

If you only have 2 data points, it is not possible to know what the relationship is between them, i.e. what the curve they lie on looks like.  The graph below illustrates this: based on different functions, the 2 data points could have a linear relationship or an exponential relationship.

*Thank you graphsketch.com
**When I took physics 1, it really annoyed me that they hadn't give us examples in calculus, because it made it so clear.  Memorizing formulas without context is a waste of time.