A logarithm is the power to which a number is raised get another number. For example, take the equation 10^{2} = 100; The superscript “2” here can be expressed as an exponent (10^{2} = 100) or as a base 10 logarithm.
For instance, the (base 10) logarithm of 100 is the number of times you’d have to multiply 10 by itself to get 100.
As it happens, this is 2, a nice neat number. But when we do logs we don’t only work with whole integers. The base 10 logarithm of 6, for instance, is 0.7781. That tells us that 10^{0.7781} equals 6.
We can summarize this all by saying that logarithms are the exponents to which a fixed number must be raised to give a certain result. The fixed number which takes the exponent is called the base. We write log_{b}x to denote the logarithm of x to base b, and this means the number that b must be raised to to get x.
Sometimes log x is written with no subscript. This is always because the writer thought it was obvious which base was meant, and the base must be inferred either from the specific context or from the general discipline.
Special logarithms
The most commonly used bases for logarithms are 10, e, and 2.
The base 10 logarithm, log_{10}x, tells us what exponent 10 has to be raised to make a given number. Log_{10} is called the common logarithm, and it is often used in engineering and science applications.
The logarithm with the number e at its base is called the natural logarithm and is often abbreviated ln; so log_{e} x = ln x. Remember that e is a special irrational number which we can’t write as a finite decimal; the first few digits are 2.718. The natural logarithm is important in both math and physics.
The base 2 logarithm is called the binary logarithm, and is often used in computer science. It is sometimes abbreviated lb.
Properties of Logarithms
Some special properties of logarithms make them very easy to use in computation. Here are the basic logarithmic laws:
- log_{b} mn = log_{b}m + log _{b}n
- log_{b}(m/n )= log_{b}m – log_{b}n
- log _{b} n^{p} = p log_{b}n
- log _{b} n = log_{a} n log_{b} a
Logarithms in Statistics
Now and then you’ll come across a logarithm or two in stats, although they aren’t widely used. If you’ve worked with logarithms before (perhaps in algebra), you may remember having to rearrange logs and solve some pretty complex equations like log_{2}(x) + log_{2}(x-2) = 3. You probably won’t see these types of equations in elementary statistics, but you might see the occasional use of a log like log^{2}.
In more advanced statistical analysis, logistic regression and Cox regression use logarithmic coefficients. Some distributions, like the reciprocal distribution or the lognormal distribution, use logarithms in their pdfs.
The Logarithm Function
The logarithm function, f(x) = log_{b}x is the inverse of the exponential function. These functions are used occasionally in error measurement, especially when using least squares to fit a linear forecasting model to logged data. A model’s error statistics fitted to natural-logged data are approximate measures of percentage error (Nau, n.d.).
Examples of Logarithms in Real Life
The Richter scale is a logarithmic scale. It is also one of the best examples of how graphs (and statistics) can lie. Why? You’ve probably heard that when an earthquake struck Haiti in 2010 was a 7.0 on the Richter scale or that the Great Japan Earthquake of 2011 was a 9.0. But did you know that the Japan earthquake was about one hundred times more powerful than the earthquake in Haiti? That’s difficult to digest, judging by that two point jump.
Magnitude
The Richter scale is on a scale of -2 (the smallest) to 9 (the largest). The reason for those huge jumps in magnitude between each digit is that the Richter scale is logarithmic. Each one digit jump in the Richter scale means roughly a ten-fold increase in ground movement and about thirty-fold increase in energy release. Therefore, it’s difficult (or impossible) to visualize the difference between a, say, 5 and 8 magnitude earthquake.
Richter Scale Chart
In the following chart, one erg is equal to 10^{−7} joules.
Richter Scale(Energy Released in millions of ergs)
- -2 (600) 100 watt light bulb left on for a week
- -1 (20000) Smallest earthquake detected at Parkfield, CA
- 0 (600000) Seismic waves from one pound of explosives
- 1 (20000000) A two-ton truck traveling 75 miles per hour
- 2 (600000000)
- 3 (20000000000) Smallest earthquakes commonly felt
- 4 (600000000000) Seismic waves from 1,000 tons of explosives
- 5 (20000000000000)
- 6 (600000000000000)
- 7 (20000000000000000) 1989 Loma Prieta ,CA earthquake (magnitude 7.1)
- 8 (600000000000000000) 1906 San Francisco earthquake (magnitude 8.3)
- 9 (20000000000000000000) Largest recorded earthquake (magnitude 9.5)
Can we predict earthquakes?
Yes, and no. Earthquake statistics help scientists to predict where and when an earthquake might take place. However, predictions are only possible when there is adequate historical data —and a lot of it. That means there are a few, well-studied areas (like Parkfield, CA) where scientist can make somewhat accurate predictions about where and when earthquakes might occur. So in general, the Richter scale can’t be used to predict earthquakes.
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References
Abramowitz, M. and Stegun, I. A. (Eds.). “Logarithmic Function.” §4.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 67-69, 2003.
Beyer, W. H. “Logarithms.” CRC Standard Mathematical Tables, 31st ed. Boca Raton, FL: CRC Press, pp. 159-160 and 221, 1987.
Conway, J. H. and Guy, R. K. “Logarithms.” The Book of Numbers. New York: Springer-Verlag, pp. 248-252, 1996.
Glen, S. Logarithmic Functions. Retrieved March 11, 2029 from:
Math Review: Useful Math for Everyone. Section 4. What is a Logarithm? Retrieved from http://www.mclph.umn.edu/mathrefresh/logs.html on December 8, 2018.
Nau, R. The logarithm transformation. Retrieved 3/11/2020 from: https://people.duke.edu/~rnau/411log.htm
Pappas, T. “Earthquakes and Logarithms.” The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 20-21, 1989.
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