Powers and the exponential function
By: Adam Bi
Tags: Calculus, Mathematics, Exponential Function, Logarithm, Powers

Powers and the exponential function

This is the second article of the series. We will define the exponential function, with the help of which we could finally properly define powers of any real exponents. Readers of this article are expected to be familiar with contents covered in the previous articles of this series, including powers of rational exponents and the ln function.

In the previous article, we studied the natural logarithm function ln and its main properties. Those properties tell us that ln is a strictly increasing function that goes from -∞ to +∞, so we could see ln is a bijective function, which means it is both injective: for any x, y ∈ (0,+∞), if ln(x) = ln(y), then x = y; and surjective: for any y ∈ ℝ, there is a positive real number x such that ln(x) = y. Or in other words, there is a one-to-one relation between (0,+∞) and ℝ under ln function.

Now, given any x ∈ ℝ, from the above discussion we know there is a positive real number y satisfying ln(y) = x, because ln is surjective, and such a positive real number is unique, because ln is injective. Such a relationship could be used to define a function:

If you are familiar with the concept of inverse function, the above definition might be easier to understand. exp is defined to be the inverse function of ln. Do not fuss too much about x and y! They are just placeholders for numbers in domain or codomain of the functions we consider. Given a real number x, exp(x) is defined to be the unique positive real number y that satisfies ln(y) = x.

A function’s plot is symmetric to its inverse function’s plot with respect to the line y = x by definition. Figure 1 is the plots for ln and exp functions on one graph.

Figure 01. Plots of exp and ln functions

As for ln function, the main properties for exp function are listed below.

Properties 1 and 2 could be derived directly from the definition or just by observing the above graph. For property 3, first we assert here that exp is differentiable on ℝ, which is a result from the inverse function theorem and is out of the scope of this article. Given that, let’s consider equation ln(exp(x)) = x for x ∈ ℝ, take derivatives of both sides, by chain rule we could get

Since exp(x) is always positive, rearranging the above result completes our proof.

Proofs for properties 4 and 5 need properties of ln function we learned in the previous article, they are not too hard and are simply given below without explanation.

Again, for property 5, we need to point out that r is restricted to rational numbers only, which is inherited from the corresponding property of ln function used in the proof.

For any r ∈ ℚ, using property 5 and take x = 1, we have

which means that values of exp function at r concides with the power e^r for all rational numbers! Based on this observation, it’s natural to make the following definition

Notice that we have made a major leap towards our target. Now we have the definition of powers of e with any real exponents! There are two points to make regarding this definition. First, if x is a rational number, we showed that exp(x) is the same as e^x; if x is an irrational number, we defined e^x to be exp(x). So in general, exp(x) is equivalent to e^x for any real number x. e^x might be the exponential function you learned in high school, and we won’t distinguish between exp(x) and e^x in the future. Second, the definition for e^x for irrational x just guarantees it is a well-defined meaningful real number. If we want to calculate an approximate value of the power, we need to know the value of exp(x), which might be hard or impossible to get. Definitions like this appear a lot in modern mathematics.

Finally, we give the definition of powers with any real exponents in a similar manner.

Notice that for a positive real number b and a rational number r, we have

So powers with rational exponents could be expressed using the exponential function. Based on this result, we could simply extend the formula to irrational exponents:

The entire process is the same as the exp(x) case we just did. Finally, we could express powers with any real exponents in a consistent way:

In this article, we have solved the big problem proposed in the beginning of the series and finally properly defined powers with any real exponents (and with positive bases). But our journey is far from the end. There are a lot of interesting and important results associated with Euler’s number e that haven’t been introduced. And what about powers with negative bases? Amazingly, definition for powers with negative bases could be properly formulated only through the exploration of the exponential function in the complex field. All of these are topics for the following articles of this series!