The two bases for logarithms in common use are a) 10 and b) the transcendental number e = 2.71828----

a) For ordinary computations, logarithms to the base 10 are most common. This is the common or Briggsian system first devised by Henry Briggs (1560-1631) with the assistance of Napier.

b) Logarithms to the base e are called Naperian or natural logarithms for their inventor John Napier (1550-1617). For general numerical computation they have no advantage over common logarithms. Their origin and usefulness must be sought in more sophisticated mathematics than just multiplication and division of numbers and is connected to the special properties of the number e. Some of these properties are:

 1. The exponential function ex is the only one that has itself as a derivative i.e.,
 

dex/dx = ex

2. The simple but very important differential equation expressing constant percentage growth with time t
 

dx/x = (const)dt

has the solution
 

x = x0e(const)t

 or taking natural logarithms

 ln(x/x0) = (const)t

3. The numerical values of logarithms are determined by summing the terms of an infinite series. The series for e is

e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ----

[Remember n! is n-factorial or (n)X(n-1)X(n-2)X---(1)]

 For e raised to the power x it is
 

ex = 1 + x + x2/2! + x3/3! + ---

Evaluating numbers in this way is labour intensive so series that converge rapidly are the best. The most rapidly converging series is that for ln (i.e., to the base e). Any other base logarithm (e.g., to base b) can be found from the simple relation
 

logba = ln a/ln b

Therefore once the most efficient set of logarithms have been found any others follow from a simple ratio.
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