# Chapter 5. Introduction of Western Mathematics

## Column Logarithm Table (Level 2)

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

We know that ax×ay=ax+y; and when ax=p, ay=q, we define the function loga that maps p to x such that p↦x(where x satisfies ax=p). So, we can write logap=x,logaq=y, and if loga(pq)=z, then z=x+y because az=pq=axay=ax+y. Consequently, if we calculate logam 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 logam and that of logan, adding them, and finding the sum in the table because loga(pq)=logap+logaq 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 10x=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.

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