log vs linear scales - John Authers (Bloomberg)

Let’s Go Logarithmic

I received a question from a reader on log scales, and I think it is worth answering. Here it is:

(BBO) Arizona Covid Debate Exposes Our Loss of Trust: John Authers                                                                                                                             
It puzzles me as to how and where it was decided the proper way to interpret the data visually is with graphs using logarithmic scales.  The general assumption I think when seeing a "graph"  
picture is that it is linear in scale.  In my opinion the  graphs are there to simplify - but they are made complicated by using a scale most people are not familiar with.  I think it would be great if someone explained to us all why that scale works better for helping to explain the problems.
 

OK, log scales are probably less well understood now than they used to be, because the advent of calculators has meant that the art of doing complex calculations using log tables is now dead. I was probably in the last generation that spent a couple of years in middle school with a book of log tables on their desk at every math (or maths as we called it in England) lesson.

A logarithmic scale is set so that percentage increases look identical. On a linear scale, the gap between 4 and 8 is twice the gap between 2 and 4. On a log scale, the gap is the same. In both cases we have seen a doubling.

Log scales  become very useful when measuring the spread of a pandemic, because a virus spreads exponentially. If everyone who catches it infects one other person, it will double. What matters crucially is to reduce its rate of change. If not many have lost their lives yet, yet cases are doubling from a low level, that is very bad news. Conversely, the same number of deaths later in the spread of the disease can be good news (in terms of the risk of future spread) because it shows that the rate of spread has changed. 

To show the phenomenon at work, here are New York coronavirus cases on both a linear (right hand side) and log scale. On the log scale, we can see that the rise in cases was deeply alarming in early March, even though the total number of cases was very low. We can also see that the spread of the disease was largely under control by the end of April, even though it would still affect many more people:

All the references to “flattening the curve,” one of the phrases we never want to hear again after Covid-19, are to curves plotted on log and not linear scales.

The other great use of log scales is for covering very long time periods. Below, we have the S&P 500 since 1927, on a price basis, on both a linear and a log scale. On a linear scale, it looks as though virtually all the gains over the last century came in the last decade. The Black Monday crash of 1987 is just barely visible; the Great Crash, the Great Depression and the ensuing recovery all appear as a straight line.

In a case like this, I think it is obvious that a log scale is far more useful. You can look at it and tell at any given point how well you would have done to invest there. Looking at it this way also helps us maintain a little more calm about the last few months. On a linear scale, the crash from February into March looks by far the worst in history. It wasn’t.

I hope this was useful. I like questions like this, because I can answer them. This doesn’t apply to that very popular question: “Will the market go up or down?” Feel free to ask more.