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Difference between revisions of "Change of base formula"

(New page: The change of base formula, shown below, is a property of logarithms. It states that for any positive <math>d,a,b</math> such that none of <math>d,a,b</math> are <math>1</math>, we have: ...)
 
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The change of base formula, shown below, is a property of logarithms. It states that for any positive <math>d,a,b</math> such that none of <math>d,a,b</math> are <math>1</math>, we have:
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The '''change of base formula''' is a formula for expressing a [[logarithm]] in one base in terms of logarithms in other bases.
  
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<cmath>\log_a b = \frac{\log_d b}{\log_d a}</cmath>
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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
  
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It is called the change of base formula because <math>\log_a b</math> can be expressed as a quotient of logarithms of any base <math>d</math>. For example, when approximating logarithms with calculators, we use the change of base formula when <math>d=e</math> so we can use the natural log key on the calculator.
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<cmath>\log_b a = \frac{\log_d a}{\log_d b}.</cmath>
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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
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<cmath>\log_b a \cdot \log_d b = \log_d a.</cmath>
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=== Use for computations ===
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The change of base formula is useful for simplifying certain computations involving logarithms.  For example, we have by the change of base formula that
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<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>
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=== Special cases and consequences ===
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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>.
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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>.
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{{stub}}
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[[Category:Elementary Algebra]]

Revision as of 13:51, 21 July 2009

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 numbers $d,a,b$ such that neither $d$ nor $b$ are $1$, we have

\[\log_b a = \frac{\log_d a}{\log_d b}.\]

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

\[\log_b a \cdot \log_d b = \log_d a.\]

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

\[\log_{\frac{1}{4}} 32\sqrt{2} = \frac{\log_2 32\sqrt{2}}{\log_2 \frac{1}{4}} = \frac{\frac{11}{2}}{-2} = -\frac{11}{4}.\]


Special cases and consequences

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

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


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