### Exponential Data

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.

When the data is exponential it can quickly become difficult to plot on a linear graph ( Medium.com, accessed 30/07/2020)

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!).

Visual representation of the exponential growth in volume of ejecta and height of ash cloud for each increase in the VEI.  (Wikipedia, accessed 30/07/2020)

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.

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.

Earthquake magnitudes and energy release, a comparison with other natural and man-made events. (USGS, accessed 30/07/2020)

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.