Logarithms

< 32.9 Modular Arithmetic | Topic Index | 32.11 Spherical Geometry >

Making Multiplication Easier 

Mathematicians also found that the idea of exponentiation can be used to simplify multiplication of very large numbers.

Let us first write down the exponents of number 2 and 3.

We know that 32 X 64 = 2048. We can see that the exponents of 32, 64 &amp; 2048 to the Base 2 are 5, 6 &amp; 11. We see that 5 + 6 = 11!

We also know that 243 X 2187 = 531441. Here again the exponents of that 243, 2187 &amp; 531441 to the Base 3 are 5, 7 &amp; 12. Here again 5 + 7 = 12!

Hence the product of the numbers is that number whose exponent is the sum of the exponents of the numbers (to the same Base).

Logarithms

We can see the entire process detailed above, in another perspective, which is called Logarithm.

If 32 = 25, then 5 is called the &ldquo;logarithm&rdquo; or log of 32 to the base 2. Similarly, 6 is log of 64 to the base 2 and that the log of 2048 to the base 2 is 11

So, if we know the log of 2 numbers (32 & 64) to the same base (2), then we know that the log of their product (2048 in this case) is given by the sum of their logs (5 + 6 = 11 in this case)

Hence the idea of logarithm has enabled us to convert a multiplication process into an addition process!

The idea of Logarithms is a powerful one in math. The idea that it helps us to convert multiplications into additions, is one of the elementary ideas. The true meaning of Logarithms is beyond the scope of this book.

Slide Rule

We have talked about using 2 ordinary rulers with cm markings for performing additions & subtractions.

Assume that we have 2 meter rulers with 100 cm markings on each. To add 45 & 37, we keep the 37 mark of one of the rulers coinciding with the 0 mark on the other ruler held below it. Then at the 37 mark of the lower ruler, we can read the total as 82.

The Slide Rule used the above principle and combined it with the exponential idea which converts multiplication into addition. In this slide rule, the markings were as per the exponential values of numbers. By using two rulers in the same way gave us the product of the numbers rather than the sum of the numbers.

The Slide Rule dramatically reduced the multiplication & division computations of astronomers.

< 32.9 Modular Arithmetic | Topic Index | 32.11 Spherical Geometry >