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Logarithm Rules

Algebra rules used when working with logarithms.

 

For the following, assume that x, y, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.

 

Definitions

1. loga x = N means that aN = x.

2. log x means log10 x. All loga rules apply for log. When a logarithm is written without a base it means common logarithm.

3. ln x means loge x, where e is about 2.718. All loga rules apply for ln. When a logarithm is written "ln" it means natural logarithm.
    Note: ln x is sometimes written Ln x or LN x.

 

Rules

1. Inverse properties:   loga ax = x   and   a(loga x) = x

2. Product:  loga (xy) = loga x + loga y

3. Quotient:  

4. Power:   loga (xp) = p loga x

5. Change of base formula:

 

Careful!!

loga (x + y) ≠ loga x + loga y

loga (x – y) ≠ loga x loga y

 


  this page updated 28-jul-14
Mathwords: Terms and Formulas from Algebra I to Calculus
written, illustrated, and webmastered by Bruce Simmons
NCTM Web Bytes December 2004 Web Bytes March 2005 Web Bytes