# 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 *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_{a}*p*=*x*,log_{a}*q*=*y*, and if log_{a}(*pq*)=*z*, then *z*=*x*+*y* because *a*^{z}=*pq*=*a*^{x}*a*^{y}=*a*^{x+y}.
Consequently, if we calculate log_{a}*m* 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_{a}*m* and that of log_{a}*n*, adding them, and finding the sum in the table because log_{a}(*pq*)=log_{a}*p*+log_{a}*q* 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 log_{10}*p*=*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 10^{1}=10、10^{2}=100、10^{3}=1,000, and 10^{4}=10,000, it follows that log_{10}10=1, log_{10}100=2,log_{10}1,000=3, log_{10}10,000=4.
Let’s calculate an example using the common logarithm table.
If we want to calculate 1,034×2,213, we should find
log_{10}1,034=log_{10}(1.034×10^{3})=3+log_{10}1.034=3+0.014520539=3.014520539 and
log_{10}2,213 =log_{10}(2.213×10^{3})=3+log_{10}2.213=3+0.344981414=3.344981414, and obtain
6.3595501953 from 3.014520539+3.344981414=6.3595501953.
Next, when we look for *x* that satisfies
log_{10}*x*=0.359501953 in the table, we should find *x*=2.288242. Therefore, the answer to 1,034×2,213 is 10^{6}×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 log_{10}1,034+log_{10}2,213+log_{10}3,256+log_{10}4,378.
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*↦log_{10}sin *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.