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Naperien logarithm

The naperien logarithm is defined as the exponent of an exponential function

eln x = x



We start from the exponential function

en = x

and want to calculate n. We must answer the question: To what power do you need to raise the base e to get x ? The answer is

to the power   ln x

The spelling scares everyone off in the beginning. Let's just remember that the logarithm is the inverse of the exponential function, because

en = x  has the inverse  n = ln x

A logarithmic function itself describes a "stray" exponent.

Shown here are ex (red) and ln x (green).


Example 1

From the definition follows

eln (7) = 7

as ln (7) = 1.945910149 and so

e1.945910149 = 7



In 1614 the Scottish mathematician John Napier published his book Mirifici Logarithmorum Canonis Descriptio.

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