Earthquakes and Logarithmic Scales

I’m hoping not to hear a lot more news about today’s earthquake in Indonesia, on the grounds that no news is good news. Quite frankly, the less newsworthy this turns out to be the better. About half an hour after the event I hit on Sky News’ live coverage. They were talking about a magnitude 8.2 quake, as reported by the US Geological Survey. A few minutes later they mentioned the Associated Press report saying it was an 8.5. I think that was attributed to a Japanese meteorological organisation. A glance at Google News tells me it’s up to 8.7 now. Thankfully there’s no news (and that’s good news, remember) of a tsunami, and the death toll seems reasonably low (though it’s about 300 more than any of us wants it to be).

So in the welcome absence of additional news I figured I’d write up a brief note on the Richter magnitude scale. I thought this would be a good idea to mention when the Sky newscaster said that 8.2 isn’t much less than 9.0 (the eventual magnitude of the December quake). It is. In fact it’s over six times less or, if you prefer, it’s less than one sixth as big. The Richter scale is a logarithmic scale, so 2.0 is ten times 1.0, 3.0 is ten times 2.0, and so on. Now here’s a mistake I’ve seen with logarithms too many times to count. If 2.0 is ten times 1.0, then 1.5 is not five times 1.0. How do you figure out how much bigger 1.5 is than 1.0? Read on.

2.0 is ten times 1.0 because 102 (ten to the power of two, equal to a hundred) is tens time 101 (ten to the power of one, equal to ten). So 1.5 is 101.5 ÷ 101 or 3.2 times 1.0 on a logarithmic scale. That makes sense because 2.0 is also 3.2 times 1.5. So 2.0 is 3.2 x 3.2 = 10 times 1.0. Knowing all of this, how much bigger is a 9.0 quake than an 8.2? The same calculation (109.0 ÷ 108.2) gives us a factor of about 6.3. Note that 9.0 ÷ 8.2 is less than 1.1 so if it was a linear scale they’d be pretty much the same quake.

This large increase in size (or strength, or whatever you want to call it) with a small increase in magnitude is what lets us use the same scale to cover all earthquakes from the tiniest un-noticable tremor to the biggest quake ever recorded (Chile in 1960). The same principle is applied to sound using the decibel scale in which an increase of ten decibels is an tenfold increase in sound energy, and to the size and brightness of stars.

Of course the huge increase in strength with magnitude also means that the change in estimates of the magnitude of today’s quake from 8.2 to 8.7 represents a threefold increase in estimated strength. I continue to hope it doesn’t get revised any farther upwards. Ditto for the death toll, which stands at just under 300 as I write this.