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Change of Base Formula

Change of Base Formula

A formula that allows you to rewrite a logarithm in terms of logs written with another base. This is especially helpful when using a calculator to evaluate a log to any base other than 10 or e.

 

Assume that x, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.

Change of base formula: Formula showing log base a of x equals (log base b of x) divided by (log base b of a)

 

Example 1: log base 16 of 32 = (log base 2 of 32) / (log base 2 of 16) = 5/4

 

Example 2: log base 2 of 3 equals (log base 10 of 3)/(log base 10 of 2) ≈ 0.47712/0.30103 ≈ 1.585    (note that 2 to the power of 1.585 is approximately equal to 3)

 

Example 3: log base 8 of x equals (ln x)/(ln 8) equals (1/ln 8) times ln x

 

See also

Logarithm rules, common logarithm, natural logarithm