Now that we have a good reason to pick a particular base, we will
be talking a lot about the new function
and its
inverse function
. This function is so useful that
it has its own name,
,
the natural logarithm.
Properties of the Natural Logarithm
Properties of the logarithm and the exponential functions are directly
related to properties of simple powers which we summarize here in
terms of our favourite base e :
Properties of the exponential function




Now since the natural logarithm
, is defined specifically
as the inverse function of the exponential function,
, we have the following two identities:
From these facts and from the properties of the exponential function
listed above follow all the properties of logarithms below.
Properties of the natural logarithm




For your consideration:
 Verify each of the properties of logarithms listed
above by using only the fact that it is the inverse of
the exponential function and the elementary properties of powers.
This is an excellent way to become familiar with the logarithm.
 Use the above information to show that we can convert bases
as follows:
where
.
More generally,
In other words, if we understand the exponential function
, then we can understand every other exponential function.
 Use this to verify that
This agrees with what we found when we considered derivatives of exponentials. There we found
that the derivative of
was proportional
to . Now we have identified the constant of
proportional as being the natural logarithm of a.

The derivative of the Natural Logarithm
Now that we understand the natural logarithm, we might ask about its
rate of growth or its derivative. Since we know that exponential
functions grow rapidly, we should expect that the natural logarithm
grows slowly.
We will begin by noting that if
, it follow
that
. This expression can be differentiated
implicitly to give
which says that
In other words,
Notice that as
becomes very large, the derivative
becomes very small. This verifies our earlier statement that the rate
of growth of the natural logarithm should be small.

