# Basic Math: Logarithms

### A logarithm (of the base b) is the power to which the base needs to be raised to yield a given number.

### If x equals b raised to the power of y,

## x=b^{y}

### then the log of x (base b) equals y.

## log_{b}(x)=y

### So, for example, using a base of 10 (log_{10} or log base 10), the logarithm of 1,000 equals 3 because 10 raised to the three equals 1,000.

## If 10 raised to the power of three equals 1,000,

## 10^{3}=1,000

## then the log (base 10) of 1,000 equals 3.

## log_{10}1000=3

### The natural logarithm is one of the most commonly used logs in statistics. This logarithm has the constant e as its base ( e = approximately 2.718281828459…). The definite integral of 1 to e of 1/x dx equals one (you don’t need to remember that, just one of the elegant gems of mathematics!!!). The natural log is often written ln(x) and is equivalent to log_{e}(x). Now it’s time for the log rules.

**The Log Rules!!!**

### To manipulate logarithms, you should understand and be able to apply the following rules. Remember that ln = log base e (log_{e}).

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### This website was developed by Elizabeth A. Albright, PhD of the Nicholas School of the Environment, Duke University. Follow Elizabeth A. Albright, PhD on Twitter @enviro_prof.

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