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 $\ne 1$. That is, $0<b<1$ or $b>1$.

Common Bases

A few notational conveniences are introduced for common bases.

Natural

The natural logarithm, denoted $\ln x$, assumes a base $e$.

Oftentimes the logarithm is defined before exponentials. If $x$ is a positive real number, the natural logarithm is defined as the following Riemann integral: $$\ln{x} = \int_1^x \frac{1}{t} ,dt.$$

Common

The common logarithm, denoted $\log x$, assumes a base $10$.

Binary

The binary logarithm, denoted $\lg x$, assumes a base $2$.

Properties

Inverse Rule

Given $b>0,b\ne 1$ and $x\in \mathbb{R}$, $$\large \log_b{b^x} = x \quad\text{and}\quad b^{\log_b{x}} = x$$

Product Rule

Given $b>0$, $b\ne 1$ and $x,y>0$, $$\large \log_b{(x \cdot y)} = \log_b{x} + \log_b{y}.$$

Quotient Rule

Given $b>0$, $b\ne 1$ and $x,y>0$, $$\large \log_b{(x \div y)} = \log_b{x} - \log_b{y}.$$

Power Rule

Given $b>0$, $b\ne 1$ and $x,y>0$, $$\large \log_b{(x^y)} = y\log_b{x}.$$

Change of Base Rule

Given $p,q>0$ such that $p\ne 1$ and $q\ne 1$, and $x>0$, $$\large \log_p{x} = \frac{\log_q{x}}{\log_q{p}}.$$

Euler's Number

Euler's number, denoted by symbol $e$, is defined as the value such that $\ln e=1$. In other words, $$\large \int_1^e \frac{1}{t} ,dt = 1.$$

Bibliography

• Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
• Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).
• Umbarger, Dan. Explaining Logarithms, n.d.
• Wikipedia. “Logarithm.” June 24, 2025. https://en.wikipedia.org/w/index.php?title=Logarithm.