# Logarithm

Pronunciation: /ˈlɔg.əˌrɪð.əm/ Explain

 Click on the blue point and drag it to change the figure. A function and its inverse are reflected about the line y=x. Manipulative 1 - Logarithm as Inverse of Exponentiation Created with GeoGebra.

A logarithm is the functional inverse of exponentiation. Written in algebraic notation, Figure 1: Definition of logarithm.

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

In manipulative 1, 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 = loge x being the inverse of y = ex.

### How to Convert an Exponent to a Logarithm

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

loga y = x iff y = ax.

5x = 25 x = log5 25.

### How to Convert a Logarithm to an Exponent

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

loga y = x iff y = ax.

Start with log2 x = 3. Substitute like values into the definition.

log2 x = 3 x = 23.

### How to Use a Logarithm 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 log3 5 = 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 thing as log10 14. 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 loge x. 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

 Click on the blue points on the sliders and drag them to change the figure What happens when the base (b) is set to 1? Why? Manipulative 2 - Graph of a Logarithmic Function Created with GeoGebra.
1. McAdams, David E.. All Math Words Dictionary, logarithm. 2nd Classroom edition 20150108-4799968. pg 110. Life is a Story Problem LLC. January 8, 2015. Buy the book

• McAdams, David E.. Change of Base Formula. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 8/31/2018. http://www.allmathwords.org/en/c/changeofbaseformula.html.

McAdams, David E. Logarithm. 4/24/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/l/logarithm.html.

### Revision History

4/24/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)