III.25 The Exponential and Logarithmic Functions

1 Exponentiation

The following is a very well-known mathematical sequence: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, . . . . Each term in this sequence is twice the term before, so, for instance, 128, the seventh term in the sequence, is equal to 2 × 2 × 2 × 2 × 2 × 2 × 2. Since repeated multiplications of this kind occur throughout mathematics, it is useful to have a less cumbersome notation for them, so 2 × 2 × 2 × 2 × 2 × 2 × 2 is normally written as 2⁷, which we read as “2 to the power 7” or just “2 to the 7.” More generally, if a is any real number and m is any positive integer, then a^m stands for a × a × . . . × a, where there are m as in the product. This product is called “a to the m,” and numbers of the form a^m are called the powers of a.

The process of raising a number to a power is known as exponentiation. (The number m is called the exponent.) A fundamental fact about exponentiation is the following identity:

a^m+n = a^m · aⁿ

This says that exponentiation “turns addition into multiplication.” It is easy to see why this identity must be true if one looks at a small example and temporarily reverts to the old, cumbersome notation. For instance,

2⁷ = 2 × 2 ×2 × 2 × 2 × 2 ×2

     = (2×2 ×2) × (2× 2× 2 ×2)

= 2³ × 2⁴.                    

Suppose now that we are asked to evaluate 2^3/2. At first sight, the question seems misconceived: an essential part of the definition of 2^m that has just been given was that m was a positive integer. The idea of multiplying one-and-a-half 2s together does not make sense. However, mathematicians like to generalize, and even if we cannot immediately make sense of 2^m except when m is a positive integer, there is nothing to stop us inventing a meaning for it for a wider class of numbers.

The more natural we make our generalization, the more interesting and useful it is likely to be. And the way we make it natural is to ensure that at all costs we keep the property of “turning addition into multiplication.” This, it turns out, leaves us with only one sensible choice for what 2^3/2 should be. If the fundamental property is to be preserved, then we must have

2^3/2. 2^3/2 = 2^3/2+3/2 = 2³ = 8.

Therefore, 2^3/2 has to be ±. It turns out to be convenient to take 2^3/2 to be positive, so we define 2^3/2 to be .

A similar argument shows that 2⁰ should be defined to be 1: if we wish to keep the fundamental property, then

2 = 2¹ = 2¹⁺⁰ = 2¹ .2⁰ = 2 . 2⁰.

Dividing both sides by 2 gives the answer 2⁰ = 1.

What we are doing with these kinds of arguments is solving a functional equation, that is, an equation where the unknown is a function. So that we can see this more clearly, let us write f (t) for 2^t. The information we are given is the fundamental property f (t + u) f(t)f(u) together with one value, f(1) = 2, to get us started. From this we wish to deduce as much as we can about f.

It is a nice exercise to show that the two conditions we have placed on f determine the value of f at every rational number, at least if f is assumed to be positive. For instance, to show that f (0) should be 1, we note that f(0)f(1) = f(1), and we have already shown that f(3/2) must be . The rest of the proof is in a similar spirit to these arguments, and the conclusion is that f (p/q) must be the q th root of 2^p. More generally, the only sensible definition of a^p/q is the qth root of a^p.

We have now extracted everything we can from the functional equation, but we have made sense of a^t only if t is a rational number. Can we give a sensible definition when t is irrational? For example, what would be the most natural definition of 2? Since the functional equation alone does not determine what 2 should be, the way to answer a question like this is to look for some natural additional property that f might have that would, together with the functional equation, specify f uniquely. It turns out that there are two obvious choices, both of which work. The first is that f should be an increasing function: that is, if s is less than t, then f (s) is less than f (t). Alternatively, one can assume that f is CONTINUOUS [I.3 §5.2].

Let us see how the first property can in principle be used to work out 2. The idea is not to calculate it directly but to obtain better and better estimates. For instance, since 1.4 < < 1.5 the order property tells us that 2 should lie between 2^7/5 and 2^3/2, and in general that if p/q < < r/s then 2 should lie between 2^p/a and 2^r/S. It can be shown that if two rational numbers p / q and r/s are very close to each other, then 2^p/q and 2^r/s are also close. It follows that as we choose fractions p / q and r / s that are closer and closer together, so the resulting numbers 2^p/q and 2^r/s converge to some limit, and this limit we call 2.

2 The Exponential Function

One of the hallmarks of a truly important concept in mathematics is that it can be defined in many different but equivalent ways. The exponential function exp(x) very definitely has this property. Perhaps the most basic way to think of it, though for most purposes not the best, is that exp(x) = e^X, where e is a number whose decimal expansion begins 2.7182818. Why do we focus on this number? One property that singles it out is that if we differentiate the function exp(x) = e^x, then we obtain e^x again—and e is the only number for which that is true. Indeed, this leads to a second way of defining the exponential function: it is the only solution of the differential equation f’(x) = f (x) that satisfies the initial condition f (0) = 1.

A third way to define exp(x), and one that is often chosen in textbooks, is as the limit of a power series:

known as the Taylor series of exp(x). It is not immediately obvious that the right-hand side of this definition gives us some number raised to the power x, which is why we are using the notation exp(x) rather than e^x. However, with a bit of work one can verify that it yields the basic properties exp(x+y) = exp(x) exp(_y), exp(0) = 1, and (d/dx) exp(x) = exp(x).

There is yet another way to define the exponential function, and this one comes much closer to telling us what it really means. Suppose you wish to invest some money for ten years and are given the following choice: either you can add 100% to your investment (that is, double it) at the end of the ten years, or each year you can take whatever you have and increase it by 10%. Which would you prefer?

The second is the better investment because in the second case the interest is compounded: for instance, if you start with $100, then after a year you will have $110 and after two years you will have $121. The increase of $11 in the second year breaks down as 10% interest on the original $100 plus a further dollar, which is 10% interest on the interest earned in the first year. Under the second scheme, the amount of money you end up with is $100 times (1.1)¹⁰, since each year it multiplies by 1.1. The approximate value of (1.1)¹⁰ is 2.5937, so you will get almost $260 instead of $200.

What if you compounded your interest monthly? Instead of multiplying your investment by 1 ten times, you would multiply it by 1 120 times. By the end of ten years your $100 would have been multiplied by (1 +)¹²⁰, which is approximately 2.707. If you compounded it daily, you could increase this to approximately 2.718, which is suspiciously close to e. In fact, e can be defined as the limit, as n tends to infinity, of the number (1 + )ⁿ.

It is not instantly obvious that this expression really does tend to a limit. For any fixed power m, the limit of (1 +)^m as n tends to infinity is 1, while for any fixed n, the limit as m tends to infinity is ∞. When it comes to (1 + )ⁿ, the increase in the power just compensates for the decrease in the number 1 + and we get a limit between 2 and 3. If x is any real number, then (1 + )ⁿ also converges to a limit, and this we define to be exp(x).

Here is a sketch of an argument that shows that if we define exp (x) in this way, then we obtain the main property that we need if our definition is to be a good one, namely exp(x) exp(y) = exp(x + y). Let us take

which equals

Now the ratio of 1 + x/n + y/n + xy/n² to 1 + x/n + y/n is smaller than 1 + x/y/n², and (1 + xy/n²)ⁿ can be shown to converge to 1 (as here the increase in n is not enough to compensate for the rapid decrease in xy/n²). Therefore, for large n the number we have is very close to

Letting n tend to infinity, we deduce the result.

3 Extending the Definition to Complex Numbers

If we think of exp(x) as e^x, then the idea of generalizing the definition to complex numbers seems hopeless: our intuition tells us nothing, the functional equation does not help, and we cannot use continuity or order relations to determine it for us. However, both the power series and the compound-interest definitions can be generalized easily. If z is a complex number, then the most usual definition of exp(z) is

Setting z = iθ, for a real number θ, and splitting the resulting expression into its real and imaginary parts, we obtain

which, using the power-series expansions for cos(θ) and sin(θ), tells us that exp(iθ) = cos(θ) + isin(θ), the formula for the point with argument θ on the unit circle in the complex plane. In particular, if we take θ = π, we obtain the famous formula e^iπ = -1 (since cos(π)= -1 and sin(π =0).

This formula is so striking that one feels that it ought to hold for a good reason, rather than being a mere fact that one notices after carrying out some formal algebraic manipulations. And indeed there is a good reason. To see it, let us return to the compound-interest idea and define exp(z) to be the limit of (1 +z/n)ⁿ as n tends to infinity. Let us concentrate just on the case where z = iπ: why should (1 + iπ/n)ⁿ be close to -1 when n is very large?

To answer this, let us think geometrically. What is the effect on a complex number of multiplying it by 1 + iπ/n? On the Argand diagram this number is very close to 1 and vertically above it. Because the vertical line through 1 is tangent to the circle, this means that the number is very close indeed to a number that lies on the circle and has argument π/n (since the argument of a number on the circle is the length of the circular arc from 1 to that number, and in this case the circular arc is almost straight). Therefore, multiplication by 1 + iπ/n is very well approximated by rotation through an angle of π/n. Doing this n times results in a rotation by π, which is the same as multiplication by -1. The same argument can be used to justify the formula exp(iθ) = cos(θ) + isin(θ).

Continuing in this vein, let us see why the derivative of the exponential function is the exponential function.

We know already that exp(z + w) = exp (z) exp (w), so the derivative of exp at z is the limit as w tends to zero of exp (z) (exp (w) - 1)/w. It is therefore enough to show that exp (w) - 1 is very close to w when w is small. To get a good idea of exp(w) we should take a large n and consider (1 + w/n)ⁿ. It is not hard to prove that this is indeed close to 1 + w, but here is an informal argument instead. Suppose that you have a bank account that offers a tiny rate of interest over a year, say 0.5%. How much better would you do if you could compound this interest monthly? The answer is not very much: if the total amount of interest is very small, then the interest on the interest is negligible. This, in essence, is why (1 + w/n)ⁿ is approximately 1 + w when w is small.

One can extend the definition of the exponential function yet further. The main ingredients one needs are addition, multiplication, and the possibility of limiting arguments. So, for example, if x is an element of a BANACH ALGEBRA [III.12] A, then exp(x) makes sense. (Here, the power series definition is the easiest, though not necessarily the most enlightening.)

4 The Logarithm Function

Natural logarithms, like exponentials, can be defined in many ways. Here are three.

(i) The function log is the inverse of the function exp. That is, if t is a positive real number, then the statement u = log (t) is equivalent to the statement t = exp(u).

(it) Let t be a positive real number. Then

(iii) If |x| < 1 then log(1 + x) = x - x² +x³ - · · ·.

This defines log (t) for 0 < t < 2. If t ≥ 2 then log (t) can be defined as -log(1/t).

The most important feature of the logarithmic function is a functional equation that is the reverse of the functional equation for exp, namely log (st) = log(s) + log(t). That is, whereas exp turns addition into multiplication, log turns multiplication into addition. A more formal way of putting this is that forms a group under addition, and ₊, the set of positive real numbers, forms a group under multiplication. The function exp is an isomorphism from to ₊, and log, its inverse, is an isomorphism from ₊ to . Thus, in a sense the two groups have the same structure, and the exponential and logarithmic functions demonstrate this.

Let us use the first definition of log to see why log(st) must equal log(s) + log(t). Write s = exp(a) and t = exp(b). Then log (s) = a, log(t) b, and

log (st) = log(exp(a) exp(b))

= log(exp(a + b))

= a+b.

The result follows.

In general, the properties of log closely follow those of exp. However, there is one very important difference, which is a complication that arises when one tries to extend log to the complex numbers. At first it seems quite easy: every complex number z can be written as re^iθ for some nonnegative real number r and some θ (the modulus and argument of z, respectively). If z = re^iθ then log (z), one might think, should be log (r) + iθ (using the functional equation for log and the fact that log inverts exp). The problem with this is that θ is not uniquely determined. For instance, what is log(1)? Normally we would like to say 0, but we could, perversely, say that 1 = e^2πi and claim that log(1) = 2πi.

Because of this difficulty, there is no single best way to define the logarithmic function on the entire complex plane, even if 0, a number that does not have a logarithm however you look at it, is removed. One convention is to write z = re^iθ with r > 0 and 0 ≥ θ < 2π, which can be done in exactly one way, and then define log (z) to be log (r) + iθ. However, this function is not continuous: as you cross the positive real axis, the argument jumps by 2π and the logarithm jumps by 2πi.

Remarkably, this difficulty, far from being a blow to mathematics, is an entirely positive phenomenon that lies behind several remarkable theorems in complex analysis, such as Cauchy’s residue theorem, which allows one to evaluate very general path integrals.