List of logarithmic identities

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In mathematics, there are several logarithmic identities.

1 Algebraic identities
2 Calculus identities
3 Approximating large numbers
4 References

[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	$\sqrt[y]{x} = x^{1/y}$
$x^{\log(y)} = y^{\log(x)} \!\,$	because	$x^{\log(y)} = e^{\log(x)^{\log(y)}} = e^{\log(y)^{\log(x)}} = y^{\log(x)} \!\,$

Where $b$ , $x$ and $x y$ are positive real numbers. If $x y$ is positive, but $x$ is not, then $\log_b(x^y) = y \log_b(-x) \!\,$ .

[edit] Canceling 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).

[edit] Proof

Let $y = log a b$ .

Then $a y = b$ .

Take $log c$ on both sides: $log c a y = log c b$

Simplify and solve for $y$ : $y log c a = log c b$

$y=\frac{\log_c b}{\log_c a}$

Since $y = log a b$ , then $\log_a b=\frac{\log_c b}{\log_c a}$

This formula has several consequences:

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

$\log_{a^n} b = {{\log_a b} \over n}$

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

$\log_{a_1}b_1 \,\cdots\, \log_{a_n}b_n = \log_{a_{\pi(1)}}b_1\, \cdots\, \log_{a_{\pi(n)}}b_n, \,$

where $\scriptstyle\pi\,$ is any permutation of the subscripts 1, ..., n. For example

$\log_a w\cdot \log_b x\cdot \log_c y\cdot \log_d z = \log_d w\cdot \log_a x\cdot \log_b y\cdot \log_c z. \,$

[edit] Summation/subtraction

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

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

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

which gives the special cases:

$\log_b (a+c) = \log_b a + \log_b (1+\frac{c}{a})$

$\log_b (a-c) = \log_b a + \log_b (1-\frac{c}{a})$

Note that in practice $a$ and $c$ have to be switched on the right hand side of the equations if $c > a$ . Also note that the subtraction identity is not defined if $a = c$ since the logarithm of zero is not defined.

[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 $b^x = 0 \!\,$ . In fact, there is a vertical asymptote on the graph of $\log_b(x) \!\,$ when $x = 0$ .

[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

${d \over dx} \ln x = {1 \over x } = {1 \over x \ln e},\qquad x > 0$

${d \over dx} \log_b x = {1 \over x \ln b},\qquad b > 0, b \ne 1$

[edit] Integral definition

$\ln x = \int_1^x \frac {1}{t} dt$

[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:

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

$x^{\left [ 0 \right ]} = \log x$

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

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

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

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 9,808,357.09543 = 9,808,357 + 0.09543. We can then get 10^9,808,357 * 10^0.09543 ≈ 1.25 * 10^9,808,357.

Similarly, factorials can be approximated by summing the logarithms of the terms.

[edit] References

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