Logarithm

Pronunciation: /ˈlɔ gəˌrɪð əm/ Explain

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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,

log to the base a of y = x if and only if y = a^x.
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.

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

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.

Start with log2x = 3. Substitute like values into the definition.

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 Propertylog base b of M times N equals log base b M plus log base b N.
Quotient Propertylog base b of M divided by N equals log base b M minus log base b N.
Power Propertylog base b of M^k equals k times log base b of M
Change of Base Propertylog base b of M is equal to log base c of M divided by log base c of b.
Table 1: Properties of Logarithms

Graph of a Logarithm

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What happens when the base (b) is set to 1? Why?
Manipulative 2 - Graph of a Logarithmic Function Created with GeoGebra.

More Information

  • 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.

Cite this article as:

McAdams, David E. Logarithm. 8/31/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/l/logarithm.html.

Image Credits

Revision History

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.)

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