In high school science we mostly deal with linear data where
there is a constant growth rate, however, there are some instances where
students will see logarithmic scales and need to understand why they are used
and what that means about the spread of the data.
Logarithmic graphs and scales are used when there is a large
spread in the data. For example, when the spread ranges from 0.1 to 1,000 it
would be difficult or impractical to try and plot this on a graph with a linear
scale.
- This tutorial gives an excellent explanation as to why logarithmic graphs are used: Logarithmic and Semi-Logarithmic Scale
In Earth Science there are numerous instances where
logarithmic scales are used to display exponential data. The Volcanic
Explosivity Index (VEI) is one of these. The VEI is a relative measure of the
explosiveness of volcanic eruptions. The volume of the products, eruption cloud
height and qualitative observations (using terms ranging from
"gentle" to "mega-colossal") are used to determine the
explosivity value. So far, the largest volcanic eruptions recorded in history were
assigned the magnitude 8 on the VEI, however, it is an open-ended scale and an
eruption larger than this could be possible (although one would hope not to
witness it!).
- For an activity relating to this try: VEI scale in the Volcanic Hazards WASP STEM package
Earthquake magnitude scales are also logarithmic. With the
Richter Scale, the lower the number the less intense the earthquake and
therefore generally the less
dangerous. Each whole number increase in magnitude represents a tenfold
increase in measured amplitude. Therefore, a magnitude six earthquake has an
amplitude 10 times greater than that of a magnitude five and 100 times greater
than a magnitude four earthquake. In practice, the Moment Magnitude Scale is
used more commonly now than the Richter Scale, however, this too is a
logarithmic scale.
- For more information on earthquake magnitude read this article:Earthquake Magnitude, Energy Release, and Shaking Intensity
Most logarithmic scales seen in daily life will increase by
multiples of ten. However, the energy released during an earthquake is also displayed
on a logarithmic scale, with each whole number increase representing 32 times
more energy released.
- For a Maths based STEM activity relating to this try: Earthquake Energy in the Earthquake Engineering WASP STEM package
Radioactive decay is a topic taught in both Physics and
Earth Science (when studying how rocks are dated). A decay curve is an example
where there is a negative exponent. A decay curve illustrates the exponential
rate at which radioactive decay of a parent nuclide occurs. The half-life
(length of time it takes the parent nuclide to decrease by half) can be
calculated from the decay curve. If the percentage of remaining parent nuclide
is known, then the age of the sample can also be determined.
- To learn more about radioactive decay and for a suggested hands-on activity read the Radioactive Popcorn blog post
An example of a logarithmic graph, which you probably see on
a daily basis at the moment, is the number of COVID-19 cases globally. It is especially
important that people understand that this is usually presented as a
logarithmic graph, where the number of cases increases by a multiple of ten on
each increment. Although a country might look like it is “middle of the road”
it could have around 1,000 times less cases than a country high up in the graph
(and conversely 1,000 times more than a country towards the bottom of the
graph). For example, in the logarithmic
graph below Germany appears in the middle, however, when this data is displayed
on a linear graph, we see that actually its numbers are relatively low. Moreover,
it becomes clear why the data is usually presented on a logarithmic scale as it
is very difficult to read off a linear graph.
Logarithmic scales are incredibly useful for displaying data
which is spread over a large range, however, it is vital that anyone reading
the scale understands the increase is in multiples, so what appears to be a
small change can represent a large increase.
- Watch this short video which illustrates exponential growth
- Here is a fun song created by a Maths professor which explains exponential growth excellently: Alan AtKisson A song about Exponential Growth