Understanding Logarithms: Definition, Properties, and Practical Applications

Definition of a Logarithm

A logarithm answers the question: to what power must a base be raised to produce a given number? In this context, log refers to the common logarithm with base 10, while ln denotes the natural logarithm with base e (≈ 2.71828).

Key Properties of Logarithms

Logarithm of a Product

log_b(MN) = log_bM + log_bN

Logarithm of a Quotient

log_b(M/N) = log_bM – log_bN

Logarithm of a Power

log_bMⁿ = n · log_bM

These identities transform multiplication, division, and exponentiation into simpler addition, subtraction, and scaling operations, which is especially valuable in manual calculations and early computing.

Historically, tables of logarithms were used to perform multiplication and division by turning the operation into addition or subtraction of logarithmic values. The resulting sum or difference is then referenced back to the table to find the product or quotient.

Slide rules embody this principle: they employ logarithmic scales so that moving the cursor a linear distance corresponds to a logarithmic change in value. Adding or subtracting these distances on the scale yields the product or quotient of the numbers represented.

Because the marks on a slide rule’s scales follow a logarithmic progression, a straight‑line measurement of distance translates into a non‑linear relationship on the printed scale. This feature makes slide rules a powerful analog tool for engineers and scientists before the digital era.

Most slide rules also incorporated specialized scales for trigonometric functions, exponentiation, root extraction, and other frequently used mathematical operations.

RELATED WORKSHEETS:

• Logarithms for Analog Circuits Worksheet