Change of Base Formula

Pronunciation: /tʃeɪndʒ ʌv beɪs/ Explain

The change of base formula is used to change the base of logarithms.[1] The change of base formula is log base a of x = (log base b of x )/(log base b of a)..

When logarithms were first invented, there were no calculators. Mathematicians and astronomers used books published with tables of logarithms. Each table had exactly one base. To find a logarithm with another base, they used the change of base formula.

Today, most calculators and math based computer programs have logarithms in two bases: base 10 and base e. To find a logarithm in any other base, one must use the change of base formula.

Mnemonic What is this?

The base (a)  on the bottom stays on the bottom. The argument (x) stays on the top. The new base is b.
Figure 1: Change of base mnemonic.


  1. Solve log24.
    log base 2 of 4 = ( log base 10 of 4 ) divided by ( log base 10 of 2 ) which is approximately 0.602060/0.301030 which is approximately 2.
    Check 2^2 = 4
  2. Solve log522.
    log base 5 of 22 = ( natural log of 22 ) divided by ( natural log of 5 ) which is approximately 3.091042/1.609438 which is approximately 1.920573.
    Check 5^1.920573 is approximately 22.00001 which is approximately 22.


  1. Bettinger, Alvin K. and Englund, John A.. Algebra and Trigonometry. pg 113-114. International Textbook Company. January 1963. Last Accessed 8/6/2018. Buy the book

More Information

  • McAdams, David E.. Logarithm. All Math Words Encyclopedia. Life is a Story Problem LLC. 6/27/2018.

Cite this article as:

McAdams, David E. Change of Base Formula. 6/25/2018. All Math Words Encyclopedia. Life is a Story Problem LLC.

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Revision History

6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/11/2010: Added "References", "More Information". (McAdams, David E.)
11/25/2008: Initial version. (McAdams, David E.)

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