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:

log_a y = x iff y = a^x.

Start with 5^x = 25. Substitute like values into the definition.

5^x = 25 ⇒ x = log₅ 25.

How to Convert a Logarithm to an Exponent

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

log_a y = x iff y = a^x.

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 3^x = 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: 3^x = 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 log_e x. Natural logarithms are used in mathematics and science.

Properties of Logarithms

Properties of Logarithms

Name	Equation
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.

References

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

More Information

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

Cite this article as:

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

Image Credits

• All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

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.)
8/31/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
8/7/2018: Changed vocabulary links to WORDLINK format. (McAdams, David E.)
5/21/2010: Added definition of antilogarithm. (McAdams, David E.)
12/19/2009: Added "References". (McAdams, David E.)
11/25/2008: Changed manipulative so that fixed objects could not be dragged. Added change of base formula to more information. Added manipulative for graph of a logarithm (McAdams, David E.)
9/17/2008: Initial version. (McAdams, David E.)