What is a logarithmic scale? Why is it more useful for some things than a regular numerical scale?
In: Mathematics
Quick calculus lesson:
Lots of things like population rates, net worth, contagion spread, fire, technology, and others don’t “grow” or “gain popularity” etc. at a linear rate, instead they grow at an exponential rate.
Linear growth means that the *rate* at which something is changing is constant: for any given change on the x-axis, there is the same proportion of change on the y-axis. On day one (for example) you might see one unit of change. Then you have one more the next day, then one more, and so on.
Exponential growth means that the rate at which something is changing is actually increasing: in the first day (for example) you can have two units of growth. The next you might have four, then eight, then sixteen, and so on. As things spread to other things, there are more things to do the spreading.
A logarithmic scale is useful because mathematically speaking, a logarithm is the inverse of an exponent – it “cancels it out”, in a way. That is to say, if you plot an exponential function on a linear scale, you get the parabola that you’d expect – this is because the rate of change is always increasing. If you plot an exponential function on a logarithmic scale, the y-axis increases exponentially, and “‘cancels out” the exponential property of the line. This makes it look like a straight line (assuming a basic exponential function).
It’s useful because lots of things are measured better by their rate of change, not so much by their change itself. Like if you’re driving a car, you usually don’t really care how many miles you’ve gone, you’re more concerned with how many miles *per hour* you’re going right now. You are concerned about the rate of change of miles rather than just the miles.
You could take it one step further and measure your acceleration, which measures the rate of change of your miles per hour which would be a second level measure of miles (displacement). Mathematically this would be known as the second derivative of displacement (with velocity being the first derivative). If you were accelerating, then the rate of change (speed) of your miles would also be increasing, and not staying constant.
With something that grows exponentially like a virus, we don’t get much information about how many people are getting infected, or even how many people got infected during a period of time. We’re expecting it to grow exponentially so while total infections or infections in the last x unit of time are good for scaring people, we actually learn from the acceleration of the virus.
If you plot an exponentially growing virus on a logarithmic scale, it cancels out that exponential growth into a straight line. If the line is steep, it means it’s accelerating more quickly at a constant rate. If it’s shallow, then it’s accelerating more slowly at a constant rate. If the line curves up, the acceleration itself is increasing, and if it curves down, the acceleration itself is decreasing – and that tells us that either a. it’s running its course (in that there are not enough potential receivers to match potential spreaders) b. anti-viral measures are working or c. a little of both.
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