# Logarithm

Log tables can be seen in almost all practical books and copies. These tables were actually made to ease the calculation process by John Napier.

He also invented numbering rods which are now called “Napier’s Bones”.

Napier’s Bones : Wolfram alpha

(While making it, I saw a pattern in the 9th multiplication column. (18 27 36 45, 54 63 72 81) But as we go further the pattern vanishes.)

He published his work on Numbering rods in 1617 in a book named “Rabdologia”. 3 years before, in 1614 he had published his first works on logarithm in the book “Description of the Marvellous Canon of Logarithms”. It’s believed that his works were influenced by some previous methods, like Gelosia method of multiplication and method of prosthaphaeresis, which converted the multiplication of trigonometric functions into addition and subtraction.

In 1891 Genaille showed a different form of Napier’s bones named Genaille-Lucas rulers. It included rulers for each digit from $0$ to $9$. Later a set of rulers for division were also invented.

Henry Briggs was greatly interested in astronomy. His work on eclipses involved long calculations. He had already published two types of tables to aid calculation. After reading Napier’s works on logarithm Briggs wrote to his friend,

… wholly employed about the noble invention of logarithms, then lately discovered. Napper, lord of Markinston, hath set my head and hands a work with his new and admirable logarithms. I hope to see him this summer, if it please God, for I never saw a book which pleased me better or made me more wonder.

Henry Briggs published his Arithmetica logarithmica (The Arithmetic of Logarithm) in 1624. This work introduced the terms “mantissa” and “characteristic”.

Their ideas were refined later by Newton, Euler, John Wallis and Johann Bernoulli towards the end of the 17th century.

When you use base $10$, calculation becomes easier, e.g.

$\log_{10} 120 = \log_{10} (1.2)(10^2) = 2 + 0.07918$

$\log_{10} 1200 =3 + 0.07918$

$\log_{10} 12000=4 + 0.07918$

So numbers which only differ by power of $10$ have same fractional part (mantissa) and only the integer part (characteristic) is different.

I will now try to give answers to some questions, which used to confuse me in my school days. One of them was, why log becomes “$e$” in some operations?

Let’s take, $2^{a} = 4$

If we want to evaluate $a$ we can use,  $\log_2 4 = a$

Similarly if we use exponential “$e$”,  $e^{a} = b$

We get $\log_e b = a$

So if you have $\log_e b = a$ you can write $e^{a} = b$

Another question was, why we have to multiply $2.3026$ to convert natural log into base 10 log?

Using the formula to change the base of logarithm, this question can also be answered.

$\log_b x = (\log_k x)/ (\log_k b)$

If $b = 10$ and $k = e$,

$\log_{10} x = (\log_e x)/ (\log_e 10)$

$\log_e x = (\log_e 10) (\log_{10} x)$

$\log_e x = (2.3026) (\log_{10} x)$

Word length sentences are made to remember the digits of $e$ and π. Here is one of them for $e$,

“To express e, remember to memorize a sentence to simplify this.” (By John L. Greene, Beverly Hills, California).

I made a sentence in Hindi which expresses the numbers, 2.718281828…

सोचो,

गुरुत्वाकर्षण के परिणामस्वरूप बल,

प्रकाशसंश्लेषण के परिणामस्वरूप ऊर्जा,

गम्भीरतापूर्वक…

A beautiful article on John Napier and his works published by Vigyan Prasar: John Napier: The Inventor of Logarithms