Logarithmic identities - Definition and Overview

What follows is a list of identities that are useful when dealing with logarithms. All of these are valid for all positive real numbers a, b and c — except that the base of a logarithm may never be 1.

Logarithmic and exponential functions are inverses

<math>a^{\log_a b} = b<math>

<math>\log_a a^r = r \quad \mbox{for all real numbers } r<math>

These are used to solve equations in which the unknowns occur in the exponent.

Special values

<math>\log_a 1 = 0<math>

<math>\log_a a = 1<math>

Multiplication, division and exponentiation

<math>\log_c ab = \log_c a + \log_c b<math>

<math>\log_c{a \over b} = \log_c a - \log_c b<math>

<math>\log_c a^r = r\log_c a<math> for all real numbers <math>r<math>

These three identities lead to the use of logarithm tables and slide rules; knowing the logarithm of two numbers allows you to multiply and divide them quickly, as well as compute powers and roots.

Change-of-base formula

<math>\log_a b = {\log_c b \over \log_c a}<math>

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:

<math>\log_a b = \frac{1}{\log_b a}<math>

<math>\log_{a^n} b = \frac{1}{n} \log_a b<math>

<math>a^{\log_b c} = c^{\log_b a}<math>

Limits

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

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

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

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

<math>\lim_{x \to 0^+} x^b \log_a x = 0<math>

<math>\lim_{x \to \infty} {1 \over x^b} \log_a x = 0<math>

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

Derivatives of logarithmic functions

<math>{d \over dx} \log_a x = {1 \over x \ln a} = {\log_a e \over x }<math>

Integrals of logarithmic functions

<math>\int \log_a x \, dx = x(\log_a x - \log_a e) + C<math>

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

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

where <math>H_n<math> is the nth harmonic number. So, for example, the first few are:

<math>x^{\left [ 0 \right ]} = \log x<math>

<math>x^{\left [ 1 \right ]} = x \log(x) - x<math>

<math>x^{\left [ 2 \right ]} = x^2 \log(x) - \begin{matrix} \frac{1}{2} \end{matrix} \, x^2<math>

<math>x^{\left [ 3 \right ]} = x^3 \log(x) - \begin{matrix} \frac{3}{4} \end{matrix} \, x^3<math>

Then,

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

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