# Origin of Logarithmic Functions

1. Apr 17, 2013

### cmcraes

Ive repeatedly asked people and my teachers how to solve logs by hand, and ive always got the same answer "They used tables and/or taylor series" but when i ask how the tables were made no one seems to know. So i am curious as to how John Napier and other mathematicians at the time found the values for their log tables. Was it just repeated trial and error? Is their some repeated algorithm? (other then those damn taylor series Hahaha)

Also how would i calculate BY HAND: 10^k
if we assume K is irrational and/or trancendental? thanks a ton!

2. Apr 19, 2013

### Mandelbroth

How do we calculate logarithms? Well, there are various definitions we can use. For convenience, I'll use the natural logarithm as an example.

Values of the principle branch of the natural logarithm are given by
$$ln(x)=\lim_{n\rightarrow\infty}n(x^{1/n}-1)=2\sum_{k=0}^{\infty}\frac{(x-1)^{2k+1}}{(2k+1)(x+1)^{2k+1}}=\int_{1}^{x}\frac{dt}{t}$$

I would not use a Taylor series, as the Taylor series for logarithmic functions tend not to converge very nicely.

As for how to calculate $10^k$ for irrational k, consider the following example.

$$10^\pi=10^{3.1415...}=10^{3+.1+.04+.001+.0005+...}=10^3\cdot 10^{.1}\cdot 10^{.04}\cdot 10^{.001}\cdot 10^{.0005}\cdot (...)$$

3. Apr 19, 2013

### slider142

John Napier created his tables at a time when none of these analytic methods were in common use. His method is extremely tedious and very historically limited (the concept of rational exponents was not even invented yet! Logarithms were one of the key ideas that led to the consistent theory of real number exponents). He spent twenty years compiling his first tables. The method and its subsequent modifications can be found in detail in the first and second chapters of the book "e: The Story of a Number", but you can read his transliterated paper here.

Last edited: Apr 19, 2013
4. Apr 21, 2013

### cmcraes

Thanks helped a ton! Could you show me how to derive this formula? Also how does it differ from say, common log? Thanks!

5. Apr 21, 2013

### cmcraes

Looking over now i see the limit definition makes sense its just the inverse of the limit definition of e correct?