Understanding Logarithms

            “During the periods of John Napier (http://en.wikipedia.org/wiki/John_Napier), he was called a magician for inventing easier ways to multiply. But now, the calculators and computers are the actual magicians for humans!!”

Introduction

            As the above quote says, the history behind logarithms is interesting and first we need to understand the problem which made mathematicians to think about using shorter ways. Mathematicians are seriously lazy people like us. :)

  Napier had been working on his invention of logarithms for twenty years before he published his results, and trigonometry tables (sin, cos table) was the origin of his ideas at about 1594.He wanted to simplify geometric progression by tabulating it and simplifying multiplications using additions. With the additive property of powers as the concept behind his approach (2^1 * 2^2 = 2^3), he started working out an approach to find a series for easy multiplication of 0.999. It was extended then to 10 in future.

10^1/2 = sqrt(10) = 3.162277 which means log(3.162277) = ½. Similarly, We can  continue for rational powers easily, as 10^3/4 = 10^1/2*10^1/4 = sqrt(10) * sqrt(sqrt(10)) = sqrt(31.62277) = 5.623413. which means log(5.623413) = ¾. I need to learn how sqrt values moves decimal points. J But this is how logs were calculated.

            This means that recording powers of a base present inside a huge number is what logarithms do.

More info

So simply, logarithms is the record of power of a given value,

log ₂8 = 3 which means 2^3 = 8. Logarithmic tables recorded the powers of huge decimals with high precision and preparing such tables is tedious job!!

Below is the logarithmic properties and b is the base.

1. log_b(xy) = log_bx + log_by.
2. log_b(x/y) = log_bx - log_by.
3. log_b(xⁿ) = n log_bx.          
4. log_bx = log_ax / log_ab.     

Example:

Log₂ 8 = log₂4 + log₂2

Log₂ 2³ = 3*log₂2

Some diagrams from MathFun: