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The Natural Logarithm

The common logarithm is used in many scientific applications, such as the Richter scale. In mathematics, the most commonly used base for logarithms is the number e = 2.718281828...

loge is abbreviated to ln (read "lin") though Matlab uses log for this function.

Thus, by definition y = ex  if and only if  ln y = x

Replacing y in the second equation by ex we get  

The natural log of e to the x is x.



y equals e to the x
    is equivalent to
the natural log of y equals x.


Replacing x in the first equation by ln x we obtain

e to the natural log of y equals y.




It is common to write ex = exp(x).

The functions ln and exp are inverses of one another. That is, they cancel each other out.

Note   ex > 0   for any x.
Consequently,  ln x  is defined only for x>0.

Here are the graphs of ln and exp:

As y equals e to the x tends to negative infinity it approaches zero and as x tends to positive infinity the function tends to positive infinity.

Each graph is a reflection of the other with respect to the line y = x.


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