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=log_b(x) if and only if x=b^y

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. Thus,

y=log_e(x) is written y=ln(x)

Show properties of logarithms

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There are some properties of logarithmic functions that may be helpful for solving problems with logs.

• log_bb^x = x
• log_bb = 1
• log_b1 = 0
• b^log_bX = X

• Earthquakes, including magnitude scales and the Gutenberg-Richter relation
• Floods and flooding
• Grain sizes/sedimentology
• Radioactive decay and dating
• Population growth
• Changes in atmospheric CO₂

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.
  
  

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  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.

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  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=10^x) 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

  L=10^M

  Thus a plot of magnitude and frequency of earthquakes plots as a straight line on a log-normal graph.



• Log-log graphical representation:
  Hjulstrom's diagram

  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 "exponent and equal" 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.

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.
• Computers
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.

• 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.

• 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.