Derivatives of Exponentials
Rates of Growth
UBC Calculus Online Course Notes

Natural exponents and logarithms

Now that we have a good reason to pick a particular base, we will be talking a lot about the new function $ e^x $ and its inverse function $ log_e(x) $ . This function is so useful that it has its own name, $ \ln(x) $ , 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
$ e^{x+c}=e^x ~e^c $
$ e^{bx}=(e^b)^x = (e^x)^c $
$ e^0=1 $
$ e^{-bx}=\frac{1}{e^{bx}} $

Now since the natural logarithm $ ln(x) $ , is defined specifically as the inverse function of the exponential function, $ e^{x} $ , we have the following two identities:


\begin{eqnarray*} 
\ln(e^x) & = & x \\ 
e^{\ln(x)} & = & x 
\end{eqnarray*}

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
$ \ln(ax)=\ln(a)+\ln(x) $
$ \ln(a^x)=x \ln a     $
$ \ln(1)=0            $
$ \ln(1/x)=-\ln(x)     $


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: $ y=2^x=e^{cx} $ where $ c=\ln(2) $ . More generally,

    \[ 
a^x = e^{\ln(a) x} 
 \]

    In other words, if we understand the exponential function $  e^x 
 $ , then we can understand every other exponential function.

  • Use this to verify that

    \[  \frac{d}{dx}a^x = \ln(a) a^x. 
 \]

    This agrees with what we found when we considered derivatives of exponentials. There we found that the derivative of $  a^x  $ 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 $  y = \ln(x) $ , it follow that $  e^y = x  $ . This expression can be differentiated implicitly to give $  e^y y^\prime = 1  $ which says that $  y^\prime = \frac{1}{e^y} = \frac{1}{x}. $ In other words,

\[ 
\frac{d}{dx} \ln(x) = \frac{1}{x} 
 \]

Notice that as $  x  $ 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.