*earthquakes, floods and flooding, grain sizes/sedimentology, radioactive decay, population growth, changes in atmospheric CO
*

**Geologic context:**_{2}, decibel scale, pH scale

# Teaching logarithms (logs)

^{by Dr. Eric M. Baer, Geology Program, Highline Community College
}

*Jump down to:*Teaching strategies | Materials & Exercises | Student Resources

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.

**log**) is defined by

**y=log**if and only if_{b}(x)**x=b**^{y}

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=log**_{10}(x)

*(~2.71...), the logarithmic function is called the "natural logarithm" and notated as*

**e****ln**. Thus,

**y=log**is written_{e}(x)**y=ln(x)**

## Teaching Strategies: Ideas from Math Education

### Put quantitative concepts in context

- Earthquakes, including magnitude scales and the Gutenberg-Richter relation
- Floods and flooding
- Grain sizes/sedimentology
- Radioactive decay and dating
- Population growth
- Changes in atmospheric CO
_{2}

### 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

Log of earthquake frequency vs. magnitude. Details

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=10
^{x}) will plot as a line on a log-normal (also called semi-logarithmic) graph. According to the Gutenburg-Richter relationship, the frequency between earthquakes (L) and the magnitude of earthquakes (M) is related by**L=10**^{M}

- Log-log graphical representation:
**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**3**^{2}=9 is the same as 2=log_{3}9

**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
- Computers

*e*. In addition, many problems in logarithms are easier to solve in one's head than typing into a calculator correctly.

### Work in groups to do multiple day, in-depth problems

## 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 CO_{2}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.

## Student resources

- The University of Guelph Physics Department has [link http://www3.physics.uoguelph.ca/tutorials/LOG/ 'a student tutorial'] about logs.
- Purplemath has a detailed logs page with links to other topics.