List of logarithmic identities

(Redirected from Logarithmic identities)

In mathematics, there are several logarithmic identities.

1 Algebraic identities
2 Calculus identities
3 Approximating large numbers
4 Complex logarithm identities
5 Powers

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), one must calculate log₁₀(3) / log₁₀(2) (or ln(3)/ln(2), which yields the same result).

[ 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:

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

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

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

$- \log_a b = \log_a \left({1 \over b}\right) = \log_{1 \over a} b$

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

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

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

[ 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.

More generally:

$\log _{b}\sum\limits_{i=0}^{N}{a_{i}}=\log _{b}a_{0}+\log _{b}\left( 1+\sum\limits_{i=1}^{N}{\frac{a_{i}}{a_{0}}} \right)=\log _{b}a_{0}+\log _{b}\left( 1+\sum\limits_{i=1}^{N}{b^{\left( \log _{b}a_{i}-\log _{b}a_{0} \right)}} \right)$

where $a_{0},\cdots ,a_{N}>0$ .

[ Calculus identities

[ 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".

[ Derivatives of logarithmic functions

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

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

[ Integral definition

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

[ 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,

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

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

[ 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.

[ Complex logarithm identities

The complex logarithm is the complex number analogue of the logarithm function. No single valued function on the complex plane can satisfy the normal rules for logarithms. However a multivalued function can be defined which satisfies most of the identities. It is usual to consider this as a function defined on a Riemann surface. A single valued version called the principal value of the logarithm can be defined which is discontinuous on the negative x axis and equals the multivalued version on a single branch cut.

[ Definitions

The convention will be used here that a capital first letter is used for the principal value of functions and the lower case version refers to the multivalued function. The single valued version of definitions and identities is always given first followed by a separate section for the multiple valued versions.

ln(r) is the standard natural logarithm of the real number r.

Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (-π, π].