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  • Write the definition of the natural logarithm as an integral.
  • Recognize the derivative of the natural logarithm.
  • Integrate functions involving the natural logarithmic function.
  • Define the number e through an integral.
  • Recognize the derivative and integral of the exponential function.
  • Prove properties of logarithms and exponential functions using integrals.
  • Express general logarithmic and exponential functions in terms of natural logarithms and exponentials.

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 e 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 e , or any of the integration and differentiation formulas associated with these functions. By the end of the section, we will have studied these concepts in a mathematically rigorous way (and we will see they are consistent with the concepts we learned earlier).

We begin the section by defining the natural logarithm in terms of an integral. This definition forms the foundation for the section. From this definition, we derive differentiation formulas, define the number e , and expand these concepts to logarithms and exponential functions of any base.

The natural logarithm as an integral

Recall the power rule for integrals:

x n d x = x n + 1 n + 1 + C , n 1 .

Clearly, this does not work when n = −1 , as it would force us to divide by zero. So, what do we do with 1 x d x ? Recall from the Fundamental Theorem of Calculus that 1 x 1 t d t is an antiderivative of 1 / x . Therefore, we can make the following definition.

Definition

For x > 0 , define the natural logarithm function by

ln x = 1 x 1 t d t .

For x > 1 , this is just the area under the curve y = 1 / t from 1 to x . For x < 1 , we have 1 x 1 t d t = x 1 1 t d t , so in this case it is the negative of the area under the curve from x to 1 (see the following figure).

This figure has two graphs. The first is the curve y=1/t. It is decreasing and in the first quadrant. Under the curve is a shaded area. The area is bounded to the left at x=1. The area is labeled “area=lnx”. The second graph is the same curve y=1/t. It has shaded area under the curve bounded to the right by x=1. It is labeled “area=-lnx”.
(a) When x > 1 , the natural logarithm is the area under the curve y = 1 / t from 1 to x . (b) When x < 1 , the natural logarithm is the negative of the area under the curve from x to 1 .

Notice that ln 1 = 0 . Furthermore, the function y = 1 / t > 0 for x > 0 . Therefore, by the properties of integrals, it is clear that ln x is increasing for x > 0 .

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 x > 0 , the derivative of the natural logarithm is given by

d d x ln x = 1 x .

Corollary to the derivative of the natural logarithm

The function ln x is differentiable; therefore, it is continuous.

A graph of ln x is shown in [link] . Notice that it is continuous throughout its domain of ( 0 , ) .

This figure is a graph. It is an increasing curve labeled f(x)=lnx. The curve is increasing with the y-axis as an asymptote. The curve intersects the x-axis at x=1.
The graph of f ( x ) = ln x shows that it is a continuous function.

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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