Logarithm & John Napier John Napier

 A logarithm is the exponent given to a reference number (called the base) in order to write a number in ‘shorthand’. For example, if we write 102 = 100 where 2 is the exponent and 10 is the base, we say that log 100 = 2 wirh respect to base 10. Two European mathematicians, John Napier  and Henry Briggs  worked out that logarithms could be used to simplify calculations in physics and astronomy. There is good evidence that tables of the logarithms of sines and cosines were in use by the Arab astronomer, Ibn Jounis as early as the 13th century. Napier had been working on his invention of logarithms for twenty years before he published his results, and this would place the origin of his ideas at about 1594. He had been thinking of the sequences which had been published now and then of successive powers of a given number. In such sequences it was obvious that sums and differences of indices of the powers corresponded to products and quotients of the powers themselves; but a sequence of integral powers of a base, such as 2, could not be used for computations because the large gaps between successive terms made interpolation too inaccurate. So to keep the terms of a geometric progression of INTEGRAL powers of a given number close together it was necessary to take as the given number something quite close to 1.  Napier therefore chose to use 1 - 10^(-7) or 0.9999999 as his given number. To achieve a balance and to avoid decimals, Napier multiplied each power by 10^7. That is, if N = 10^7[1 - 1/10^7]^L, then L is Napier's logarithm of the number N. Thus his logarithm of 10^7 is 0.  At first he called his power indices "artificial numbers", but later he made up the compound of the two Greek wordLogos (ratio) and arithmos (number).Napier did not think of a base for his system, but nevertheless his tables were compiled through repeated multiplications, equivalent to powers of 0.9999999   Obviously the number decreases as the index or logarithm increases. This is to been  expected because he was essentially using a base which is less than 1.

Convert a decimal number into any base system from binary to Base 89.

Decimal Number:

Base:

Number in new base:

Count:

Value of any number raised to any power (xy)
--X
--Y
result

Log Calculator to any base

Logarithm of

Base:
E PI 2 10 Other

Solution: