earthquakes, floods and flooding
, grain sizes/sedimentology, radioactive decay
, population growth
, changes in atmospheric CO2
, decibel scale, pH scale
Teaching logarithms (logs)by Dr. Eric M. Baer, Geology Program, Highline Community College
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Logarithms are the inverse of the exponential function. Originally developed as a way to convert multiplication and division problems to addition and subtraction problems before the invention of calculators, logarithms are now used to solve exponential equations and to deal with numbers that extend from very large to small in a more elegant fashion. For more information on exponential functions, go to the Exponential Growth and Decay page.
The logarithm function (log
) is defined by
- y=logb(x) if and only if x=by
and x>0, b>0 and b not equal to 1.
The function is read
and is read "y is the log base b of x"
When no base (b) is noted, the assumed base is 10. Thus,
- y=log(x) is the same as y=log10(x)
When the base is the number e
(~2.71...), the logarithmic function is called the "natural logarithm" and notated as ln
- y=loge(x) is written y=ln(x)
There are some properties of logarithmic functions that may be helpful for solving problems with logs.
- logbbx = x
- logbb = 1
- logb1 = 0
- blogbX = X
Teaching Strategies: Ideas from Math Education
Put quantitative concepts in context
There are a number of geologic contexts in which logs are used and can be introduced. Some of these include:
Use multiple representations
Because everyone has different ways of learning, mathematicians have defined a number of ways that quantitative concepts can be represented to individuals. In the geosciences, logarithms are most commonly represented graphically.
- Graphical representation:
Earthquake frequency vs. magnitude. Details
One of the most common ways that geoscientists use logarithms is to plot data on a logarithmic scale. This is used when the values on a graph span large values. An example is the number of earthquakes per year of various magnitudes, plotted to the right. The values of the individual data points are unreadable because of the arithmetic scale.
Log of earthquake frequency vs. magnitude. Details
Using a logarithmic scale allows closer estimation of many of the values. However, this graph may be more difficult for students to interpret because most authors of logarithmic graphs plot data on a logarithmic scale but label the axis with non-log numbers. Students will need to be warned about this because they will often not realize that the scale is logarithmic. To find the actual location or value of a point on a logarithmic graph, the log of the original number must be calculated. This makes it very difficult for students to figure out the exact value of points, plot additional points, and can even cause misinterpretation - for example on this graph, students might think that there is a linear relationship between the frequency and magnitude of earthquakes when there is not.
Earthquake frequency vs. magnitude
Rarely is the axis clearly and correctly marked with the log values, as it is in this graph.
- Log-normal graphical representation:
Many graphical representations of logarithms are constructed with only one axis being logarithmic. A exponential function (y=10x
) will plot as a line on a log-normal (also called semi-logarithmic) graph.
The Gutenburg-Richter relationship on a log-normal graph
According to the Gutenburg-Richter relationship, the frequency between earthquakes (L) and the magnitude of earthquakes (M) is related by
Thus a plot of magnitude and frequency of earthquakes plots as a straight line on a log-normal graph.
- Log-log graphical representation:
Some graphs have two logarithmic axes. These are often called "log-log graphs." A good example is Hjulstrom's diagram which shows the stream velocities at which sediments will be eroded, transported and deposited.
- Algebraic/numerical representation:
The secret to logs is getting the algebraic representation down, so that one can convert between the log function and the exponential function. Students will need assistance in remembering this pattern. As an example, I show students
- 32=9 is the same as 2=log39
I think the use of colored chalk when writing these numbers helps. Some students also find the phrase "ex
ponent and equ
al" helpful in remembering this relationship since the exponent (2 in the above example) is followed by the equals sign when converting it to the logarithmic notation.
Use technology appropriately
Students have any number of technological tools that they can use to better understand quantitative concepts -- from the calculators in their backpacks to the computers in their dorm rooms. Logarithms can make use of these tools to help the students understand this often difficult concept.
- Graphing calculators
Graphing calculators are an easy way for all students to enter data and to see what a curve of that data looks like. All graphing calculators are slightly different and students may need help with their particular model. There are some helpful hints for some calculators at Prentice-Hall's Calculator help website (more info)
. Note that very few calculators will calculate logs in bases other than 10 or e
. In addition, many problems in logarithms are easier to solve in one's head than typing into a calculator correctly.
Logarithms provide an excellent opening for an introduction to the use of spreadsheet programs. Students are likely to encounter spreadsheet programs in many of their classes and they are excellent tools for visualizing the shape of an equation. Excel has an option to switch to logarithmic axes which can be useful for quickly illustrating the usefulness of logarithms in the analysis of data.
Work in groups to do multiple day, in-depth problems
Mathematicians also indicate that students learn quantitative concepts better when they work in groups and revisit a concept on more than one day. Therefore, when discussing quantitative concepts in entry-level geoscience courses, have students discuss or practice the concepts together. Also, make sure that you either include problems that may be extended over more than one class period or revisit the concept on numerous occasions.
Logarithms are a concept that comes up over and over in introductory geoscience: radioactive decay, Richter magnitude, pH scale, etc. When each new topic is introduced, make sure to point out that they have seen this type of function before and should recognize it.
Teaching Materials and Exercises
- Determining Earthquake Probability and Recurrence
A homework/classroom activity where students collect historical earthquake information and use it to forecast the probability of larger earthquakes.
- Using functions in an introductory geoscience course
A template and set of exercises designed to help faculty increase the graphical literacy of their students. Two exercises are included - population growth and atmospheric CO2 increase - for help in teaching exponential growth and decay. The template gives general guidelines for teaching students the relationship between functions and their graphical representation.
- Scaling Galileo's Solar System - Size of the Globes
In this module, students determine the sizes of the various planets in the solar system scaled such that the orbit of Saturn fits on campus. The students also compare the planet sizes, given the scale, to the grain sizes of different sediment types. It includes plotting on logarithmic axes in excel.
- Two streams, two stories... How Humans Alter Floods and Streams
Students plot stream flow data on logarithmic axes in order examine flooding.