List of logarithmic identities

In mathematics, there are several logarithmic identities.

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

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).

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).

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. \, $$

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.

Trivial identities
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 $$x=0$$.

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.

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 logb(a) + logb(c) = logb(a*c), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 232,582,657 - 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 109,808,357 * 100.09543 ≈ 1.25 * 109,808,357.

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