Logarithms

Overview

In the equation $b^{y} = x$ , the exponent $y$ is known as the logarithm of the number $x$ using a base $b$ , denoted $y = \log_{b} x$ . The characteristic refers to the integer part of the logarithm whereas the mantissa refers to the fractional part. The mantissa is always a positive number between $0$ and $1$ .

In real analysis, base $b$ is restricted to a positive value $\neq 1$ . That is, $0 < b < 1$ or $b > 1$ .

Common Bases

A few notational conveniences are introduced for common bases.

The natural logarithm, denoted $\ln x$ , assumes a base $e$ .
The common logarithm, denoted $\log x$ , assumes a base $10$ .
The binary logarithm, denoted $\lg x$ , assumes a base $2$ .

Definition

If $x$ is a positive real number, the natural logarithm is defined as the following Riemann integral:

\ln x = \int_{1}^{x} \frac{1}{t} d t .

This is often generalized to accommodate any nonzero $x \in R$ as:

\ln | x | = \int_{1}^{| x |} \frac{1}{t} d t .

Properties

Inverse Rule

Given $b > 0, b \neq 1$ and $x \in R$ ,

\log_{b} b^{x} = x and b^{\log_{b} x} = x

Product Rule

Given $b > 0$ , $b \neq 1$ and $x, y > 0$ ,

\log_{b} (x \cdot y) = \log_{b} x + \log_{b} y .

Quotient Rule

Given $b > 0$ , $b \neq 1$ and $x, y > 0$ ,

\log_{b} (x \div y) = \log_{b} x - \log_{b} y .

Power Rule

Given $b > 0$ , $b \neq 1$ and $x, y > 0$ ,

\log_{b} (x^{y}) = y \log_{b} x .

Change of Base Rule

Given $p, q > 0$ such that $p \neq 1$ and $q \neq 1$ , and $x > 0$ ,

\log_{p} x = \frac{\log_{q} x}{\log_{q} p} .

Euler's Number

Euler's number, denoted by symbol $e$ , is defined as the value such that $\ln e = 1$ . In other words,

\int_{1}^{e} \frac{1}{t} d t = 1.