## What are Logarithms?

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:

**The base ten logarithm of 100 (written as log_{10} 100) is 2, because 10^{2} = 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 log

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

Another way to think of the word *log *is that it’s a question. If you see the phrase log_{10} 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 10
^{3}= 10,000 - Log 1000 = 3, because 10
^{3}= 1000 - Log 100 = 2, because 10
^{2}= 100 - Log 10 = 1, because 10
^{1}= 10

## Bases and Arguments

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.

## Examples in base *e*

- Log
_{e}5 = 1.6094379124, because e^{1.6094379124}= 5 - Log
_{e}100 = 4.605170186, because e^{4.605170186}= 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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## Logarithmic vs. Exponential Formulas

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 formula example: log
_{a}x = y - Exponential formula example: a
^{y}= x

## 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.

## 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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