﻿ Logarithm: The value of an exponent, the inverse of exponentiation.

# Logarithm

Pronunciation: /ˈlɔ gəˌrɪð əm/ ?

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A logarithm is the functional inverse of exponentiation.[1] Written in algebraic notation,

 Figure 1: Definition of logarithm.

The exponent, x in the equation, is also called the antilogarithm.

Click on the blue point in manipulative 1 and drag it to change the figure. Note that the value of x and y on the point on the blue line (y=ex), is swapped on the point on the green line (y=logex). This is a consequence of y=logex being the inverse of y=ex.

### Converting Exponents to Logarithms

To convert an exponent to a logarithm, use the definition of logarithms:

logay = x iff y = ax.

5x = 25 x = log525.

### Converting Logarithms to Exponents

To convert a logarithm to an exponent, use the definition of logarithms:

logay = x iff y = ax.

log2x = 3 x = 23.

### Using Logarithms to Solve a Problem

An equation such as 3x = 5 is difficult to solve without using logarithms. However, converting the equation to a logarithm makes it easy to solve. Use the definition of a logarithm: 3x = 5 log35 = x. Calculating the logarithm gives x ≈ 1.46497.

### Common Logarithm

The common logarithm is a logarithm using a base 10. When writing a common logarithm, leave out the base: log 14 means the same things as log1014. Common logarithms are commonly used in business applications.

### Natural Logarithm

The natural logarithm is a logarithm with a base e. The natural logarithm is written ln x. ln x means the same thing as logex. Natural logarithms are used in mathematics and science.

### Properties of Logarithms

Properties of Logarithms
NameEquation
Product Property
Quotient Property
Power Property
Change of Base Property
Table 1: Properties of Logarithms

### Graph of a Logarithm

 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Manipulative 1: Graph of a Logarithm. Created with GeoGebra.

• McAdams, David. Change of Base Formula. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Change%20of%20Base%20Formula.
• McAdams, David. Inverse of a Function. allmathwords.org. Life is a Story Problem LLC. 2009-03-12. http://www.allmathwords.org/article.aspx?lang=en&id=Inverse%20of%20a%20Function.

Logarithm. 2010-05-21. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/l/logarithm.html.