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

Powers and the exponential function

This is the third article of the series. We will introduce exponentials and logarithms with other bases from what we have learned so far. Readers should be familiar with the natural logarithm and exponential functions, they were introduced in the previous articles.

Before introducing exponentials and logarithms with other bases, let’s complement some of our previous results to make them more powerful.

Consider a power function f: (0,+∞) → ℝ f(x) = x^a, where a ∈ ℝ. Until now, we assume (implicitly) that we know the derivative of this function when a is a rational number, which could be easily proved by the definition of differentiation. In fact, we have to use the following result to prove the property 5 of the ln function.

Since we already know the meaning of the power for all real exponent, it is time to generalise the above result. Remember f(x) = x^a = exp(aln(x)), by chain rule, we get

We are happy to see the rule for a power’s derivative still holds even if the exponent of the power is irrational. Besides, the proof also shows that as for the rational case, if a is irrational, we still have x^(a - 1) = x^a/x for any positive x.

The following two results generalize the fifth properties of ln and exp functions, their proofs are exactly the same as before, except that we need the above derivative result to handle irrational r.

As ln is called the natural logarithm function, the exp function we studied in the previous article is often referred as the natural exponential function. Exponential functions with other bases are defined based on it:

Notice that an exponential function with base b is different from a power function with exponent b. A power function has its independent variable x at its base while an exponential function has its independent variable x at its exponent. In addition, we eliminate 1 in the definition for nothing special but the fact that 1 to the power of anything still equals 1, which gives a boring function. 1^x = exp(xln(1)) = exp(0) = 1.

According to our definition, b^x = exp(xln(b)) = exp(t), which could be regarded as a stretch or compression of exp function. As we noted previously , exp is a bijective function, then so is b^x, which guarantees that we could define its inverse function.

We have drawn the graphs of several exponential and logarithm functions with different bases in Figure 1.

Figure 01. Exponential and logarithm funtions with different bases

Based on the above definition, the inverse function of e^x is the logarithm function of base e. But we also know e^x = exp(x) for any real x, and the inverse of exp is ln, so ln is the logarithm function of base e by definition. Many results could be further deduced for exponential and logarithm functions, you might have learned most of them in high school, and we won’t dive into them further to prevent us from unnecessary digression. The only result we will introduce about them is the following change of bases property.

By now, we have introduced many definitions, the most important ones of which is summarized in Figure 2. We started from the definition of ln function using fundamental theorem of calculus, and exp is defined to be the inverse of ln. exp is then used to define powers of any real exponents. Finally, the inverse of exponential functions of general bases are defined to be logarithm functions of general bases.

Figure 02. Logic of key definitions

The definitions we have learned are quite useful. For example, let’s consider a famous limit result.

The base of the power approaches to 1 as x approaches to +∞, it is trying to “drag” the value of the power to 1. While the exponent of the power approaches to +∞ as x approaches to +∞, and it is trying to “drag” the value of the power to infinity. What will be the combined result of these two forces? With the help of our definition and L’Hôpital’s rule, the result turns out to be e!

Until now we have mostly finished what we want to say about logarithm functions. We will focus more on the exponential function in the future. In the next article, our main task is to study the Taylor series of exp function, which could help us to prove e is irrational and find an approximation of it with any precision!