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Logarithms

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...

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

Thus, by definition y = e^x ln y = x

Replacing y in the second equation by e^x we get

Replacing x in the first equation by ln x we obtain

It is common to write e^x = exp(x).

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

Note e^x > 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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