If you were asked to calculate 4199710.4828 + 2.1829765, you could probably finish it in less than a minute (including verification). But what if it was 4199710.4828 Ă 2.1829765? Or even more dauntingly, 4199710.4828^2.1829765? Youâd probably start frantically using a calculator. If you werenât allowed to use a calculator, youâd surely protest, because this would be too much! And if the calculation were an unimaginable 41997104.8282^1.829765, youâd probably wonder, "Are the question setters insane?" But did you know? Long before the invention of calculators, European mathematicians and astronomers faced countless challenging computational problemsâproblems involving arithmetic operations and power and root operations similar to the aforementioned numbers. These tedious calculations cost countless people countless hours and effort. So naturally, someone wondered: Could there be a way to improve calculations? Finally, someone came up with a brilliant solution: the invention of logarithms.
Logarithms are the inverse of exponential operations. Napier, the inventor of logarithms, studied two series of numbers. When the first series increased in an arithmetic progression and the second series increased in a geometric progression, he established a simple correspondence between the product of each pair of numbers in the second series and the sum of the corresponding two numbers in the first series. This laid the foundation for the idea of logarithms. After years of contemplation, Napier published his work, and logarithms began to spread. Many mathematicians worked tirelessly to compile logarithm tables, making indelible contributions to human civilization. With logarithms, multiplication and division could be transformed into addition and subtraction. For example, when calculating 9710.3265Ă·2.1829 (known as the true numbers), we take their logarithms, look up lg9710.3265 = 3.9872 and lg2.1829 = 0.3390 in the logarithm table, subtract them to get 3.6482, and then look up the inverse logarithm table to obtain the value of 9710.3265Ă·2.1829, which is 4448.3607. The above value is an approximation, but as long as the logarithm table is accurate enough, the error can be minimized, making it sufficiently accurate for most calculations. Similarly, logarithms can convert exponentiation into multiplication: lg(ab) = blga. If you still find multiplication too complex, you can further transform it into addition using logarithms and then look up the inverse logarithm table until you obtain the correct solution.
Therefore, the invention of logarithms was indeed a major innovation in mathematical methods, a significant improvement for the increasingly demanding scientific calculations of the time. Some might say that with electronic computers today, people no longer need to refer to logarithm tables for complex calculations. But donât forget that logarithms and the logarithmic function are also a major creation of mathematics itself, ubiquitous in mathematical theory and natural sciences. As the famous French mathematician Laplace said, "The discovery of logarithms extended the lifespan of astronomers." Galileo once exclaimed, "Give me space, time, and logarithms, and I can create a universe!"