- Understand the natural logarithm and the mathematical constant e using integrals
- Identify how to differentiate the natural logarithm function
- Perform integrations where the natural logarithm is involved
- Understand how to find derivatives and integrals of exponential functions
- Convert logarithmic and exponential expressions to base e forms
We already examined exponential functions and logarithms in earlier chapters. However, we glossed over some key details in the previous discussions. For example, we did not study how to treat exponential functions with exponents that are irrational. The definition of the number is another area where the previous development was somewhat incomplete. We now have the tools to deal with these concepts in a more mathematically rigorous way, and we do so in this section.
For purposes of this section, assume we have not yet defined the natural logarithm, the number , or any of the integration and differentiation formulas associated with these functions.
The Natural Logarithm as an Integral
Recall the power rule for integrals:
This rule doesn’t work for as it would force us to divide by zero. So, how do we integrate According to the Fundamental Theorem of Calculus, we know that is an antiderivative of
Therefore, we define the natural logarithm function as follows:
defining the natural logarithm
For define the natural logarithm function by
For this represents the area under the curve from to For we have,
,
so it is the negative of the area under the curve from .

Notice that Furthermore, since for , is increasing for .
Properties of the Natural Logarithm
Because of the way we defined the natural logarithm, the following differentiation formula falls out immediately as a result of to the Fundamental Theorem of Calculus.
derivative of the natural logarithm
For the derivative of the natural logarithm is given by
The function is differentiable; therefore, it is continuous.
A graph of is shown below. Notice that it is continuous throughout its domain of

Calculate the following derivatives:
Note that if we use the absolute value function and create a new function we can extend the domain of the natural logarithm to include Then This gives rise to the familiar integration formula.
integral of (1/) du
The natural logarithm is the antiderivative of the function
Calculate the integral
Although we have called our function a “logarithm,” we have not actually proved that any of the properties of logarithms hold for this function. We do so here.
properties of the natural logarithm
If and is a rational number, then
Proof
- By definition,
- We have
Use on the last integral in this expression. Let Then Furthermore, when and when So we get
- Note that
Furthermore,
Since the derivatives of these two functions are the same, by the Fundamental Theorem of Calculus, they must differ by a constant. So we have
for some constant Taking we get
Thus and the proof is complete. Note that we can extend this property to irrational values of later in this section.
Part iii. follows from parts ii. and iv. and the proof is left to you.
Use properties of logarithms to simplify the following expression into a single logarithm: