Powers and the exponential function
This is the first article of the entire series. We will formally define natural logarithm function and Euler’s number e, with the help of which we could further define powers with any real exponents. Readers of this article are expected to be familiar with basic operations of real numbers (addition, subtraction, multiplication and division) and some calculus results.
At the end of the previous article of this series, we threw out the problem of defining powers with any real exponents, or in particular, with irrational exponents like 3^sqrt(2). Our answer to this question starts with the definition of a function you might already be familiar with, that is the ln function. With the annoying barrier I have made for us, that is, we don’t know how to define powers with irrational exponents, you might find out your old way of defining ln function could no longer be used. Luckily, there are other alternative definitions of ln. And the one we will use requires fundamental theorem of calculus:
We won’t discuss too much about this theorem. Our target is using it to define ln function. The important idea from this theorem is that given a continuous function (f), a differentiable function (F)could be defined by taking its integration, and f is just the derivative of F.
Now, it’s easy to see f: (0,+∞)→ℝ, f(x)=1/x is a continuous function, so the theorem guarantees that we could make the following definition:
Here we define ln(x) to be the integration of function f(t) = 1/t from 1 to x, x ∈ (0,+∞). According to the theorem, we could directly know ln is differentiable in its domain (and hence continuous of course), with (ln(x))’ = f(x) = 1/x for all x ∈ (0,+∞). Notice that we could plug in any positive real number as independent variable x to obtain ln(x) ∈ ℝ. Though we might not know the exact decimal form of ln(x), we know it is a meaningful real number, which is vouched by fundamental theorem of calculus. Figure 1 is the graph of ln function.
Our version of ln is exactly the same function you might already learned in high school. Many of its properties you are familiar with could still be proved based on this definition, they are listed below.
Properties 1 and 2 tell us ln is a strictly increasing function that goes from -∞ to +∞, with 1 being its only zero point. Proofs of these two properties require knowledge of Riemann integral, which is out of the scope of this article. Keeping in mind of the graph of ln is probably the best way to understand these two properties.
Properties 3 to 5 provide three useful techniques for calculations involving ln. Notice that under ln function, multiplication and division could be transformed into addition and subtraction. This is actually one of the motivations for introducing logarithm in the first place. Proofs for properties 3 to 5 are similar, we will only show the proof for property 3 below.
Notice that in property 5, r is restricted to ℚ instead of ℝ. This is the furthest we could get at the current stage, since we have only properly defined powers with rational exponents. And remember, defining powers with any real exponents is our main target. How ln function could help us achieve this target will be revealed gradually in next article.
Since ln monotically goes from -∞ to +∞, we know there is a unique positive real number x_0 satisfying ln(x_0) = 1. Hence it’s meaningful to make the following definition:
e is the famous Euler’s number, which is widely used in science and engineering. There are also several other methods to define this number, and we will encounter one of which later in this series. In one of the future article, we will also prove that e is an irrational numer, and e ≈ 2.71828.
With the definition of Euler’s number e, we conclude this article. In the next article of this series, we will define the exponential function as the inverse function of ln, with the help of which we could finally define powers with any real exponents!
