Logarithmic identities

From Wikipedia, the free encyclopedia

It has been suggested that Change of base formula for logs be merged into this article or section. (Discuss)

In mathematics, there are several logarithmic identities.

1 Algebraic identities
2 Calculus identities
3 Approximating large numbers

[edit] Algebraic identities

[edit] Using simpler operations

Logarithms can be used to make calculations easier. For example, two numbers can be multiplied just by using a logarithm table and adding.

$\log_b(xy) = \log_b(x) + \log_b(y) \!\,$	because	$b^x \cdot b^y = b^{x + y} \!\,$
$\log_b\!\left(\begin{matrix}\frac{x}{y}\end{matrix}\right) = \log_b(x) - \log_b(y)$	because	$\begin{matrix}\frac{b^x}{b^y}\end{matrix} = b^{x - y}$
$\log_b(x^y) = y \log_b(x) \!\,$	because	$(b^x)^y = b^{xy} \!\,$
$\log_b\!\left(\!\sqrt[y]{x}\right) = \begin{matrix}\frac{\log_b(x)}{y}\end{matrix}$	because	$\sqrt[y]{x} = x^{1/y}$

[edit] Cancelling exponentials

Logarithms and exponentials (antilogarithms) with the same base cancel each other. This is true because logarithms and exponentials are inverse operations (just like multiplication and division).

$b^{\log_b(x)} = x$	because	$\mathrm{antilog}_b(\log_b(x)) = x \!\,$
$\log_b(b^x) = x \!\,$	because	$\log_b(\mathrm{antilog}_b(x)) = x \!\,$

[edit] Changing the base

$\log_a b = {\log_c b \over \log_c a}$

This identity is needed to evaluate logarithms on calculators. For instance, most calculators have buttons for ln and for log₁₀, but not for log₂. To find log₂(3), you have to calculate log₁₀(3) / log₁₀(2) (or ln(3)/ln(2), which is the same thing).

This formula has several consequences:

$\log_a b = \frac {1} {\log_b a}$

$\log_{a^n} b = (\frac {1} {n}) \log_a b$

$a^{\log_b c} = c^{\log_b a}$

[edit] Logarithmic "snake" identity

Any sequence

$\!\ \log_a(b) \cdot \log_b(c) \cdot \log_c(d)\cdots \log_y(z)$

can be simplified to...

$\!\ \log_a(z)$

This leads to the identity

$\log_b a = {\log_b c \over \log_a c}.\,$

[edit] Summation/subtraction

The following summation/subtraction rule is especially useful in probability theory when one is dealing with a sum of log-probabilities:

$\log_b (a+c) = \log_b a + \log_e (1+e^{\log_b c - \log_b a})$

$\log_b (a-c) = \log_b a - \log_e (1+e^{\log_b c - \log_b a})$

[edit] Trivial identities

$\log_b(1) = 0 \!\,$	because	$b^0 = 1\!\,$
$\log_b(b) = 1 \!\,$	because	$b^1 = b\!\,$

Note that: $\log_b(0) \!\,$ is undefined because there is no number $x \!\,$ such that $b^x = 0 \!\,$ .

[edit] Calculus identities

[edit] Limits

$\lim_{x \to 0^+} \log_a x = -\infty \quad \mbox{if } a > 1$

$\lim_{x \to 0^+} \log_a x = \infty \quad \mbox{if } a < 1$

$\lim_{x \to \infty} \log_a x = \infty \quad \mbox{if } a > 1$

$\lim_{x \to \infty} \log_a x = -\infty \quad \mbox{if } a < 1$

$\lim_{x \to 0^+} x^b \log_a x = 0$

$\lim_{x \to \infty} {1 \over x^b} \log_a x = 0$

The last limit is often summarized as "logarithms grow more slowly than any power or root of x".

note: to say the limit of a function "equals infinity" is not strictly correct notation, as "infinity" is not a value. What is meant by the limits equations above is simply that the functions increase/decrease without bound.

[edit] Derivatives of logarithmic functions

${d \over dx} \ln x = {1 \over x } = {\ln e \over x }$

[edit] Integral definition

${d \over dx} \log_e x = \int_1^x \frac {1}{t} dt$

[edit] Integrals of logarithmic functions

$\int \log_a x \, dx = x(\log_a x - \log_a e) + C$

To remember higher integrals, it's convenient to define:

$x^{\left [n \right]} = x^{n}(\log(x) - H_n)^{\left [ 0 \right ]} = \log x$

$x^{\left [ 1 \right ]} = x \log(x) - x$

$x^{\left [ 2 \right ]} = x^2 \log(x) - \begin{matrix} \frac{3}{2} \end{matrix} \, x^2$

$x^{\left [ 3 \right ]} = x^3 \log(x) - \begin{matrix} \frac{11}{6} \end{matrix} \, x^3$

Then,

$\frac {d}{dx} \, x^{\left [ n \right ]} = n \, x^{\left [ n-1 \right ]}$

$\int x^{\left [ n \right ]}\,dx = \frac {x^{\left [ n+1 \right ]}} {n+1} + C$

[edit] Approximating large numbers

The identities of logarithms can be used to approximate large numbers. Note that log_b(a) + log_b(c) = log_b(a*c), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2^32,582,657 - 1. To get the base-10 logarithm, we would multiply 32,582,657 by log₁₀(2), getting 9,808,056.0654 = 9,808,056 + 0.0654. We can then get 10^9,808,056 * 10^0.0654 ≈ 1.25 * 10^9,808,056.

Retrieved from "http://en.wikipedia.org../../../l/o/g/Logarithmic_identities.html"

Categories: Articles to be merged since November 2006 | Logarithms | Mathematical identities