The Natural Logarithm as an Integral

Recall the power rule for integrals:

Clearly, this does not work when [latex]n=-1,[/latex] as it would force us to divide by zero. So, what do we do with [latex]\int \frac{1}{x}dx?[/latex] Recall from the Fundamental Theorem of Calculus that [latex]{\int }_{1}^{x}\frac{1}{t}dt[/latex] is an antiderivative of [latex]1\text{/}x.[/latex] Therefore, we can make the following definition.

For [latex]x>1,[/latex] this is just the area under the curve [latex]y=1\text{/}t[/latex] from 1 to [latex]x.[/latex] For [latex]x<1,[/latex] we have [latex]{\int }_{1}^{x}\frac{1}{t}dt=\text{−}{\int }_{x}^{1}\frac{1}{t}dt,[/latex] so in this case it is the negative of the area under the curve from [latex]x\text{ to }1[/latex] (see the following figure).

Figure 1. (a) When [latex]x>1,[/latex] the natural logarithm is the area under the curve [latex]y=1\text{/}t[/latex] from [latex]1\text{ to }x.[/latex] (b) When [latex]x<1,[/latex] the natural logarithm is the negative of the area under the curve from [latex]x[/latex] to 1.

Notice that [latex]\text{ln}1=0.[/latex] Furthermore, the function [latex]y=1\text{/}t>0[/latex] for [latex]x>0.[/latex] Therefore, by the properties of integrals, it is clear that [latex]\text{ln}x[/latex] is increasing for [latex]x>0.[/latex]

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.

A graph of [latex]\text{ln}x[/latex] is shown in (Figure). Notice that it is continuous throughout its domain of [latex](0,\infty ).[/latex]

Figure 2. The graph of [latex]f(x)=\text{ln}x[/latex] shows that it is a continuous function.

Note that if we use the absolute value function and create a new function [latex]\text{ln}|x|,[/latex] we can extend the domain of the natural logarithm to include [latex]x<0.[/latex] Then [latex](d\text{/}(dx))\text{ln}|x|=1\text{/}x.[/latex] This gives rise to the familiar integration formula.

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.

Defining the Number [latex]e[/latex]

Now that we have the natural logarithm defined, we can use that function to define the number [latex]e.[/latex]

To put it another way, the area under the curve [latex]y=1\text{/}t[/latex] between [latex]t=1[/latex] and [latex]t=e[/latex] is 1 ((Figure)). The proof that such a number exists and is unique is left to you. (Hint: Use the Intermediate Value Theorem to prove existence and the fact that [latex]\text{ln}x[/latex] is increasing to prove uniqueness.)

Figure 3. The area under the curve from 1 to [latex]e[/latex] is equal to one.

The number [latex]e[/latex] can be shown to be irrational, although we won’t do so here (see the Student Project in Taylor and Maclaurin Series in the second volume of this text). Its approximate value is given by

The Exponential Function

We now turn our attention to the function [latex]{e}^{x}.[/latex] Note that the natural logarithm is one-to-one and therefore has an inverse function. For now, we denote this inverse function by [latex]\text{exp}x.[/latex] Then,

The following figure shows the graphs of [latex]\text{exp}x[/latex] and [latex]\text{ln}x.[/latex]

Figure 4. The graphs of [latex]\text{ln}x[/latex] and [latex]\text{exp}x.[/latex]

We hypothesize that [latex]\text{exp}x={e}^{x}.[/latex] For rational values of [latex]x,[/latex] this is easy to show. If [latex]x[/latex] is rational, then we have [latex]\text{ln}({e}^{x})=x\text{ln}e=x.[/latex] Thus, when [latex]x[/latex] is rational, [latex]{e}^{x}=\text{exp}x.[/latex] For irrational values of [latex]x,[/latex] we simply define [latex]{e}^{x}[/latex] as the inverse function of [latex]\text{ln}x.[/latex]

Then we have [latex]{e}^{x}=\text{exp}(x)[/latex] for all [latex]x,[/latex] and thus

for all [latex]x.[/latex]

Properties of the Exponential Function

Since the exponential function was defined in terms of an inverse function, and not in terms of a power of [latex]e,[/latex] we must verify that the usual laws of exponents hold for the function [latex]{e}^{x}.[/latex]

General Logarithmic and Exponential Functions

We close this section by looking at exponential functions and logarithms with bases other than [latex]e.[/latex] Exponential functions are functions of the form [latex]f(x)={a}^{x}.[/latex] Note that unless [latex]a=e,[/latex] we still do not have a mathematically rigorous definition of these functions for irrational exponents. Let’s rectify that here by defining the function [latex]f(x)={a}^{x}[/latex] in terms of the exponential function [latex]{e}^{x}.[/latex] We then examine logarithms with bases other than [latex]e[/latex] as inverse functions of exponential functions.

Now [latex]{a}^{x}[/latex] is defined rigorously for all values of [latex]x[/latex]. This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of [latex]r.[/latex] It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.

Let’s now apply this definition to calculate a differentiation formula for [latex]{a}^{x}.[/latex] We have

The corresponding integration formula follows immediately.

If [latex]a\ne 1,[/latex] then the function [latex]{a}^{x}[/latex] is one-to-one and has a well-defined inverse. Its inverse is denoted by [latex]{\text{log}}_{a}x.[/latex] Then,

Note that general logarithm functions can be written in terms of the natural logarithm. Let [latex]y={\text{log}}_{a}x.[/latex] Then, [latex]x={a}^{y}.[/latex] Taking the natural logarithm of both sides of this second equation, we get

Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base [latex]a.[/latex] Again, let [latex]y={\text{log}}_{a}x.[/latex] Then,