View source for Change of base formula

The '''change of base formula''' is a formula for expressing a [[logarithm]] in one base in terms of logarithms in other bases.

For any [[positive]] [[real number]]s <math>d,a,b</math> such that neither <math>d</math> nor <math>b</math> are <math>1</math>, we have

<cmath>\log_b a = \frac{\log_d a}{\log_d b}.</cmath>

This allows us to rewrite a logarithm in base <math>b</math> in terms of logarithms in any base <math>d</math>.  This formula can also be written 

<cmath>\log_b a \cdot \log_d b = \log_d a.</cmath>

=== Use for computations ===

The change of base formula is useful for simplifying certain computations involving logarithms.  For example, we have by the change of base formula that 

<cmath>\log_{\frac{1}{4}} 32\sqrt{2} = \frac{\log_2 32\sqrt{2}}{\log_2 \frac{1}{4}} = \frac{\frac{11}{2}}{-2} = -\frac{11}{4}.</cmath>


=== Special cases and consequences ===

Many other logarithm rules can be written in terms of the change of base formula.  For example, we have that <math>\log_b a = \frac{\log_a a}{\log_a b} = \frac{1}{\log_a b}</math>.  Using the second form of the change of base formula gives <math>\log_b a^n = \log_b a \cdot \log_a a^n = n \log_b a</math>.

One consequence of the change of base formula is that for positive constants <math>a, b</math>, the functions <math>f(x) = \log_a x</math> and <math>g(x) = \log_b x</math> differ by a constant factor, <math>f(x) = (\log_a b) g(x)</math> for all <math>x > 0</math>.


{{stub}}
[[Category:Elementary Algebra]]