Statistics Definitions > What are Logarithms?

## 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}. All you really need to know about logarithms used in statistics is this simple fact:

What are Logarithms? Logarithms are simply an exponent in a different form. For example log

_{a}x = y is the same as a^{y}= x.

## Logarithms: Understanding in Steps

Let’s have a look at some of the steps that make logarithms easier to understand.

Step 1: **Understand the difference between logarithmic and exponential calculations: **

This is the first and probably the easiest step of all. If you find something like log_{a}x = 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 x^{2} or a^{7}.

- Logarithmic calculation: log
_{a}x = y - Exponential calculation: a
^{y}= x

Step 2: **Understand various logarithmic parts:**

The *base *in logarithmic calculation is the *subscript* which you can find next to the letters *log *as shown in the example below. The base next to the word *log* is 3. The number following the subscript is known as the *argument *as shown in example which is number 10. Finally, the logarithmic expression is set equal to 4 in the example below.

**Sample problem:** Solve log_{2}(x) = 4

The question is telling you that a log (on the left side) equals a number (on the right side), so:

2^{4} = x

16 = x

## 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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