How a 17th-century invention turned complex multiplication into simple addition

Science · Published 20 October 2026 Swipe more like this →

In short: Before the digital age, multiplying massive numbers was a grueling, error-prone task. Logarithms revolutionized this by allowing scientists to swap out multiplication for addition. This mathematical shortcut was so efficient it served as the backbone of numerical work for over three centuries.

Logarithms are defined as the exponent to which a base must be raised to produce a specific number. For instance, because 2 raised to the power of 3 equals 8, the logarithm of 8 to the base 2 is 3. This relationship allows mathematicians to express powers in a linear format, which is essential for solving complex exponential equations and modeling various physical and biological systems.

Invented in the 17th century, logarithms were primarily designed to accelerate calculations. They were so effective that they became a fundamental tool for numerical work for more than 300 years. They only began to lose their status as a primary tool for large-scale computation in the late 19th century with the perfection of mechanical calculating machines, and later with the advent of computers in the 20th century.

The primary utility of logarithms lies in their unique properties. One of the most significant is the product rule: the logarithm of a product is equal to the sum of the logarithms of the individual numbers. This meant a scientist could find the product of two large numbers by looking up their individual logarithms in a table, adding those two values together, and then finding the 'antilogarithm' of the result. Similarly, division problems could be converted into subtraction problems.

While 'common' logarithms (base 10) were historically significant for table-based work, the natural logarithm (base e, where e is approximately 2.71828) remains a cornerstone of modern mathematics. It continues to be one of the most useful functions for creating mathematical models across the physical and biological sciences today. Mastery of these formulas remains a core component of algebra, used for expanding expressions, condensing logs, and performing change of base operations.

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