How did John Napier's logarithms simplify multiplication and division?

John Napier’s invention of logarithms revolutionized numerical calculation by transforming complex operations into simpler ones, effectively reducing the time and potential for error in tasks like astronomy and trigonometry.12

Napier recognized that multiplication and division were time-consuming and prone to "slippery errors." By utilizing the law of exponents—specifically that $a^n\times a^m=a^{n+m}$—he established a relationship between geometric series and arithmetic series.23

The simplification works as follows:
* Multiplication: Instead of multiplying two numbers ($m\times n$), one adds their corresponding logarithms: $\log (mn)=\log m+\log n$.13
* Division: Instead of dividing two numbers ($m/n$), one subtracts their logarithms: $\log (m/n)=\log m-\log n$.3

By consulting tables to find the logarithms of the numbers, performing the addition or subtraction, and then finding the antilogarithm, mathematicians could bypass direct multiplication and division entirely. This method remained a fundamental tool for large-scale computations for over 300 years.13