- What are Logarithms?
- Exponential vs. Logarithmic formulas
- Logarithms in Statistics
- Logarithms in real life: The Richter Scale
A logarithm is the power to which a number is raised get another number. For example, take the equation 102 = 100; The superscript “2” here can be expressed as an exponent (102 = 100) or as a base 10 logarithm:
The base ten logarithm of 100 (written as log10 100) is 2, because 102 = 100.
Logarithms and exponents form a symbiotic relationship—basically, they “undo” each other. To put that another way,
logarithms are simply an exponent in a different form. For example logax = y is the same as ay = x.
Another way to think of the word log is that it’s a question. If you see the phrase log10 100, it’s asking “10 raised to what power equals 100?“. When you first start learning about logs, you’ll almost always start with learning about base 10 logs (i.e. multiples of 10 like 10*10 = 100 or 10*10*10 = 1,000); in other words, if you can multiply 10 by itself, you should pick up base 10 logs pretty quickly.
Examples in base 10:
- Log 10,000 = 4, because 103 = 10,000
- Log 1000 = 3, because 103 = 1000
- Log 100 = 2, because 102 = 100
- Log 10 = 1, because 101 = 10
In a formula, the base is the subscript which you can find next to the letters log . The number following the subscript is called the argument; this is also called a power if you’re writing it in exponential form.
The base tells you the number you’re working with (i.e. the number that you’ll raise to some power). While you could technically have any number for a base, the three most common are:
- Base 10 (the decimal logarithm or common log). This is usually written as log(x),
- Base e (Euler’s number),
- Base 2 (the binary logarithm).
If no base is written, you can usually assume base 10.
- Loge 5 = 1.6094379124, because e1.6094379124 = 5
- Loge 100 = 4.605170186, because e4.605170186 = 100
Obviously, things get a little trickier when you’re dealing with base e. This calculator does the work for you:
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If you find something like logax = y then it is a logarithmic problem. Always remember logarithmic problems are always denoted by letters “log”. If the calculation is in exponential format then the variable is denoted with a power, like x2 or a7.
- Logarithmic formula example: logax = y
- Exponential formula example: ay = x
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 log2(x) + log2(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 log2.
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 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.
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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