Multiplying large numbers by hand is slow and error-prone. Adding them is not. Logarithms, introduced in the early seventeenth century, exploit a property of exponents to convert the first problem into the second: multiplying two numbers becomes, after converting to logarithms, simply adding two other numbers together and converting back. For roughly three and a half centuries before electronic calculators became commonplace, this single trick was one of the most important labour-saving devices available to anyone doing serious quantitative work.
Why multiplication turns into addition
The logarithm of a number, to a given base, is the exponent that base must be raised to in order to produce that number. The key property that makes logarithms useful for calculation follows directly from the rules of exponents: the logarithm of a product of two numbers equals the sum of each number's individual logarithm. So multiplying two large numbers can be done instead by looking up each number's logarithm in a precomputed table, adding those two logarithms together (a far easier operation than long multiplication), and then looking up which original number corresponds to that summed logarithm. John Napier, who first published on logarithms in 1614, developed the concept specifically to ease the burden of the extensive hand calculation required in fields like astronomy and navigation, where multiplying long strings of large numbers was a routine, tedious necessity.
The slide rule turned the trick into a physical tool
The slide rule, invented not long after logarithms became known, gave this mathematical trick a physical, mechanical form: two rulers marked with logarithmic scales, rather than evenly spaced linear ones, could be slid against each other so that physically adding two lengths on the logarithmic scale automatically performed the equivalent of multiplying the two original numbers, without the user needing to consult a logarithm table or do any arithmetic by hand at all. Slide rules remained the standard calculation tool for engineers, scientists, and students for over three centuries, right up until inexpensive electronic calculators displaced them in the 1970s — a striking illustration of how directly a piece of abstract mathematics, discovered to solve a specific calculation burden, translated into a physical instrument that shaped how quantitative work actually got done for generations.
What we're still unsure about
The mathematical properties of logarithms and their historical role in easing hand calculation, culminating in the slide rule, are extremely well documented, resting on both the underlying mathematics and extensive surviving historical record. What's more a matter of estimation than precise record is exactly how much scientific and engineering progress across the seventeenth through twentieth centuries was accelerated by logarithms specifically being available, as opposed to other contemporaneous calculation aids — disentangling the contribution of one specific tool from the many overlapping developments in mathematics and instrumentation over that long a period isn't something historians of science can pin down to a precise figure.
This sits inside Exponents & Logarithms, one of eight topics in Algebra, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.