Chapter 5. Introduction of Western Mathematics

The logarithm table was a numerical table developed by J. Napier (1550-1617) to convert multiplications to additions.

We know that a^x×a^y=a^x+y; and when a^x=p, a^y=q, we define the function log_a that maps p to x such that p↦x(where x satisfies a^x=p). So, we can write log_ap=x,log_aq=y, and if log_a(pq)=z, then z=x+y because a^z=pq=a^xa^y=a^x+y. Consequently, if we calculate log_am for every number m and list the results in a table, we should be able to obtain the product of m×n by finding the value of log_am and that of log_an, adding them, and finding the sum in the table because log_a(pq)=log_ap+log_aq holds true. This idea was invented to simplify calculations in the fields of navigation and astronomy, which require the calculations of numbers with many digits. The log₁₀p=x, the inverse function of 10^x=p, is called the common logarithm of p, and a table that lists the values of x and p, where x is correspondent to p, is called the table of common logarithm.

Since 10¹=10、10²=100、10³=1,000, and 10⁴=10,000, it follows that log₁₀10=1, log₁₀100=2,log₁₀1,000=3, log₁₀10,000=4. Let’s calculate an example using the common logarithm table. If we want to calculate 1,034×2,213, we should find log₁₀1,034=log₁₀(1.034×10³)=3+log₁₀1.034=3+0.014520539=3.014520539 and log₁₀2,213　=log₁₀(2.213×10³)=3+log₁₀2.213=3+0.344981414=3.344981414, and obtain 6.3595501953 from 3.014520539+3.344981414=6.3595501953. Next, when we look for x that satisfies log₁₀x=0.359501953 in the table, we should find x=2.288242. Therefore, the answer to 1,034×2,213 is 10⁶×2.2288242=2,288,242

This method is handy when we need to repeat multiplications like 1,034×2,213×3,256×4,378, because it turns such multiplications to an addition, like log₁₀1,034+log₁₀2,213+log₁₀3,256+log₁₀4,378. Particularly, to facilitate the use of logarithm calculations in astronomical calculations and surveying, which require multiplications of values of trigonometric functions with many digits, a logarithm table of a trigonometric function, x↦log₁₀sin x , has been also created. Using this table, calculations, such as sinα×sinβ×sinγ, which we often face in spherical trigonometry, can be converted into their corresponding additions.