Exponential and hyperbolic functions: synthesis and challenges

Summary

This lesson closes the unit on the exponential, logarithmic and hyperbolic functions, which is the last one of the course. It gathers its results into a single deductive chain, which starts from the completeness of the real numbers and reaches the parametrization of the hyperbola; it shows how that chain rests on the earlier units of the course, from induction and the binomial theorem to the equation of the hyperbola, and how it prepares the course Algebra and Geometry II, which begins with the trigonometric functions and polar coordinates. It ends with ten challenges, harder than the problems proposed in the lessons, which require combining several of them and finding an idea of one’s own.

Learning objectives

Reconstruct the chain of definitions and theorems of the unit, identifying in each link the earlier result on which it depends.
Relate the results of the unit to those of the earlier units and anticipate their role in the trigonometric functions of the course Algebra and Geometry II.
Solve problems that combine the monotonicity of the exponential, the laws of logarithms, the inequality $e^x \geq 1 + x$ and the hyperbolic addition formulas.
Choose on one’s own the substitution, the inequality or the auxiliary function that reduces a non-routine problem to a known one.

What the unit achieved

The unit started from a gap: at the end of the unit on number systems, the power $a^x$ made sense for rational $x$ , but not for $x = \sqrt{2}$ . Its final result is a family of functions defined on the whole line, with complete algebraic laws, explicit inverses and a geometric interpretation. The links of the chain are the following.

The power with a real exponent

The lesson on the exponential function and powers with real exponents defined $a^x$ , for $a > 1$ , as the supremum of the powers $a^r$ with $r$ rational and $r \leq x$ . Completeness guaranteed the existence of the supremum; the density of the rational numbers gave the bracketing $a^r < a^x < a^s$ for $r < x < s$ , and with it strict monotonicity; Bernoulli’s inequality bounded the length of the bracketing intervals and made it possible to prove, together with the Archimedean property, the laws of exponents, the last of them case by case according to the base and the sign of the exponent:

a^{x + y} = a^x a^y, \qquad (ab)^x = a^x b^x, \qquad \left( a^x \right)^y = a^{xy}

With these laws, the exponential function turned out to be positive, injective and strictly monotonic; a new supremum argument, together with Bernoulli’s inequality, proved that it takes every positive value, so that it is a bijection from $\mathbb{R}$ onto $\ioo{0}{\infty}$ ; and it was proved that it eventually overtakes the squares.

The number e

The lesson on the number e and the natural exponential chose a privileged base. With the binomial theorem and Bernoulli’s inequality it proved that $\left( 1 + \frac{1}{n} \right)^n$ increases and $\left( 1 + \frac{1}{n} \right)^{n + 1}$ decreases, and defined $e$ as the number that both sequences enclose; it proved that it is also the supremum of the sums of the reciprocals of the factorials, with an error smaller than $\frac{1}{n! \, n}$ , and that it is irrational, and, with the fundamental inequality and the power of a power, it proved that the supremum of the capitals $\left( 1 + \frac{r}{n} \right)^n$ is $e^r$ , without resorting to the concept of limit. The most fruitful result of the lesson is the fundamental inequality, valid for every real $x$ , with equality only at $x = 0$ , which reappears in almost all the later proofs:

e^x \geq 1 + x

The logarithm and the equations

The lesson on the logarithm function obtained $\log_a$ as the inverse of the exponential and translated each law of exponents into a law of logarithms; the change of base showed that all logarithms are multiples of a single one, and Euclid’s lemma proved that $\log_2 3$ is irrational. The lesson on exponential and logarithmic equations and inequalities turned two properties of these functions into methods: injectivity makes it possible to equate exponents and arguments, and monotonicity preserves or reverses inequalities according to the base. The difficulties proper to these problems, the domain, the extraneous solutions and the non-positive values of the auxiliary variable, all come from the fact that the exponential takes only positive values. Only one result of that lesson rests on something the unit does not prove: the existence of the solution of an equation without an algebraic method, such as $x + 2^x = 5$ , relies on the intermediate value theorem, admitted in the lesson on rational functions, and on the continuity of the exponential, which is admitted in the lesson on the power with a real exponent; the proof of both facts belongs to calculus. Its uniqueness, on the other hand, and the bisection that locates it, follow from the order.

The models and the hyperbolic functions

The lesson on growth and decay models proved that the law $P(t + h) = e^{kh} P(t)$ characterizes the model $P_0 e^{kt}$ , and that, therefore, every monotonic quantity that changes in the same proportion over equal intervals is exponential; hence the doubling time and the half-life, carbon-14 dating, Newton’s law, semi-log linearization and the logistic model. Finally, the lesson on the hyperbolic functions combined $e^x$ and $e^{-x}$ into the hyperbolic cosine and sine, whose fundamental identity is a direct consequence of the law of exponents:

\cosh^2 x - \sinh^2 x = \left( \cosh x + \sinh x \right) \left( \cosh x - \sinh x \right) = e^x e^{-x} = 1

From it followed the parametrization of the right branch of the hyperbola $x^2 - y^2 = 1$ , whose parameter is recovered with a logarithm, the addition formulas and the inverse functions, which are expressed with logarithms by solving quadratic equations in $e^y$ . The chain thus closes on itself: the exponential produces the hyperbolic functions, and the logarithm inverts them.

The following figure gathers the four segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that are admitted without proof or come from earlier units, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

ACompleteness. Every nonempty set of real numbers that is bounded above has a supremum; it was stated in the unit on number systems. Admitted without proof in this unit.
BDensity and Archimedean property. Between two distinct real numbers there is always a rational number, and every real number is exceeded by some natural number; both were proved in the unit on number systems. Admitted without proof in this unit.
CBernoulli’s inequality. If $x \geq -1$ and $n$ is a natural number, then $(1 + x)^n \geq 1 + nx$ ; proved by induction in the unit on the natural numbers. Admitted without proof in this unit.
DBinomial theorem. For every natural number $n$ we have $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n - k} y^k$ ; proved in the unit on the natural numbers. Admitted without proof in this unit.
EEuclid’s lemma. If a prime divides a product of natural numbers, it divides one of the factors; proved in the unit on the natural numbers. Admitted without proof in this unit.
FLaws of exponents. For $a > 0$ , $b > 0$ and real $x$ , $y$ we have $a^{x + y} = a^x a^y$ , $(ab)^x = a^x b^x$ and $\left( a^x \right)^y = a^{xy}$ . Established in: 3. Laws of real exponents.
GMonotonicity of the exponential. If $a > 1$ , for real numbers $u$ and $v$ we have $u < v$ if and only if $a^u < a^v$ ; with a base between $0$ and $1$ , the inequality is reversed. Established in: 2. Bracketing and monotonicity.
HLogarithm as the inverse. For $x > 0$ , $\log_a x = y$ if and only if $a^y = x$ ; hence $a^{\log_a x} = x$ and $\log_a \left( a^y \right) = y$ . Established in: 15. Logarithm.

Segment 1 The power with a real exponent
- Define $a^x$ for every real exponent as a supremum of rational powers and justify that it extends the earlier definition.
- Bracket $a^x$ between rational powers and deduce from this strict monotonicity.
- Prove the laws of real exponents, including the power of a power, with bracketings whose length is bounded by Bernoulli’s inequality.
- Describe the exponential function: domain, sign, injectivity, growth without bound and horizontal asymptote.
- Prove with completeness and Bernoulli’s inequality that the exponential takes every positive value.
1. Power with a real exponent. If $a > 1$ , the power $a^x$ is the supremum of the powers $a^r$ with $r$ rational and $r \leq x$ ; the supremum exists and, if $x$ is rational, it coincides with the power already known.
  Rests on: A (Completeness), B (Density and Archimedean property). Proved in: The exponential function and powers with real exponents.
2. Bracketing and monotonicity. If $a > 1$ and $r$ , $s$ are rational numbers with $r < x < s$ , then $a^r < a^x < a^s$ . Consequently, if $x < y$ , then $a^x < a^y$ when $a > 1$ and $a^x > a^y$ when $0 < a < 1$ ; moreover, $a^x > 0$ for every positive base.
  Uses: 1. Power with a real exponent. Rests on: B (Density and Archimedean property). Proved in: The exponential function and powers with real exponents.
3. Laws of real exponents. For $a > 0$ , $b > 0$ and real $x$ , $y$ we have $a^{x + y} = a^x a^y$ and $(ab)^x = a^x b^x$ ; with the first, the uniqueness of the principal root and a new bracketing, also $\left( a^x \right)^y = a^{xy}$ , in all the sign cases.
  Uses: 2. Bracketing and monotonicity. Rests on: B (Density and Archimedean property), C (Bernoulli’s inequality). Proved in: The exponential function and powers with real exponents.
4. Exponential function. If $a > 0$ and $a \neq 1$ , the function $a^x$ is defined on $\mathbb{R}$ and is positive and injective; if $a > 1$ , it increases, exceeds every bound and approaches $0$ to the left, and the graph of $\left( \frac{1}{a} \right)^x$ is its reflection across the $y$ axis.
  Uses: 1. Power with a real exponent. Rests on: C (Bernoulli’s inequality), G (Monotonicity of the exponential). Proved in: The exponential function and powers with real exponents.
5. Range of the exponential. If $a > 0$ and $a \neq 1$ , for every $y > 0$ there exists a unique real $x$ with $a^x = y$ : if $a > 1$ , it is the supremum of the rational numbers $r$ with $a^r < y$ . The exponential is a bijection from $\mathbb{R}$ onto $\ioo{0}{\infty}$ .
  Uses: 4. Exponential function. Rests on: A (Completeness), B (Density and Archimedean property), C (Bernoulli’s inequality), F (Laws of exponents), G (Monotonicity of the exponential). Proved in: The exponential function and powers with real exponents.
Segment 2 The number e
- Prove with the binomial theorem and Bernoulli’s inequality that $\left( 1 + \frac{1}{n} \right)^n$ increases and $\left( 1 + \frac{1}{n} \right)^{n + 1}$ decreases, and bound it with the inequality $k! \geq 2^{k - 1}$ , proved by induction.
- Define $e$ as the number that both sequences enclose.
- Prove that $e$ is the supremum of the sums of the reciprocals of the factorials, bound the error and deduce that $e$ is irrational.
- Prove the fundamental inequality $e^x \geq 1 + x$ and, with it, that the supremum of the capitals $\left( 1 + \frac{r}{n} \right)^n$ is $e^r$ .
1. Growth of the sequence. The sequence $a_n = \left( 1 + \frac{1}{n} \right)^n$ is strictly increasing.
  Rests on: D (Binomial theorem). Proved in: The number e and the natural exponential.
2. Factorials and powers of two. For every natural number $k$ we have $k! \geq 2^{k - 1}$ , with equality only if $k = 1$ or $k = 2$ ; it is proved by induction.
  Proved in: The number e and the natural exponential.
3. Bound on the sequence. For every natural number $n$ we have $a_n \leq s_n < 3$ , where $s_n = \sum_{k=0}^{n} \frac{1}{k!}$ : each term of $s_n$ is bounded by one of a geometric progression with ratio $\frac{1}{2}$ .
  Uses: 7. Factorials and powers of two. Rests on: D (Binomial theorem). Proved in: The number e and the natural exponential.
4. Companion sequence. The sequence $b_n = \left( 1 + \frac{1}{n} \right)^{n + 1}$ is strictly decreasing.
  Rests on: C (Bernoulli’s inequality). Proved in: The number e and the natural exponential.
5. The number e. The number $e$ , the supremum of the terms $a_n$ , satisfies $a_n < e < b_n$ for every natural number $n$ .
  Uses: 6. Growth of the sequence, 8. Bound on the sequence, 9. Companion sequence. Rests on: A (Completeness). Proved in: The number e and the natural exponential.
6. Series for the number e and its error. The supremum of the sums $s_n$ is $e$ , and for every natural number $n$ we have $0 < e - s_n < \frac{1}{n! \, n}$ .
  Uses: 8. Bound on the sequence, 10. The number e. Rests on: A (Completeness), B (Density and Archimedean property), C (Bernoulli’s inequality). Proved in: The number e and the natural exponential.
7. Irrationality of e. The number $e$ is irrational.
  Uses: 11. Series for the number e and its error. Proved in: The number e and the natural exponential.
8. Fundamental inequality. For every real $x$ we have $e^x \geq 1 + x$ , and equality holds only at $x = 0$ .
  Uses: 10. The number e. Rests on: B (Density and Archimedean property), C (Bernoulli’s inequality), F (Laws of exponents), G (Monotonicity of the exponential). Proved in: The number e and the natural exponential.
9. Continuous compounding. If $r > 0$ , the capitals $\left( 1 + \frac{r}{n} \right)^n$ increase with $n$ , none of them reaches $e^r$ and their supremum is $e^r$ ; for $t > 0$ , the supremum of the numbers $\left( 1 + \frac{r}{n} \right)^{nt}$ is $e^{rt}$ .
  Uses: 1. Power with a real exponent, 6. Growth of the sequence, 10. The number e, 13. Fundamental inequality. Rests on: D (Binomial theorem), F (Laws of exponents), G (Monotonicity of the exponential). Proved in: The number e and the natural exponential.
Segment 3 The logarithm and the equations
- Obtain $\log_a$ as the inverse of the exponential and translate each law of exponents into a law of logarithms.
- Reduce all logarithms to a single one by means of the change of base, and study their monotonicity and their sign.
- Prove with Euclid’s lemma that $\log_2 3$ is irrational.
- Turn injectivity and monotonicity into methods for solving equations and inequalities.
1. Logarithm. Since the exponential with base $a$ is a bijection from $\mathbb{R}$ onto $\ioo{0}{\infty}$ , for each $x > 0$ the number $\log_a x$ is the unique real $y$ such that $a^y = x$ , and the function $\log_a$ is its inverse.
  Uses: 5. Range of the exponential. Proved in: The logarithm function and its properties.
2. Laws of logarithms. For positive $x$ , $y$ and real $r$ we have $\log_a(xy) = \log_a x + \log_a y$ , $\log_a \frac{x}{y} = \log_a x - \log_a y$ and $\log_a \left( x^r \right) = r \log_a x$ .
  Rests on: F (Laws of exponents), H (Logarithm as the inverse). Proved in: The logarithm function and its properties.
3. Change of base. We have $\log_a x = \frac{\log_b x}{\log_b a}$ ; every logarithm is the natural logarithm divided by a constant, and $\log_a b \cdot \log_b a = 1$ .
  Uses: 16. Laws of logarithms. Rests on: H (Logarithm as the inverse). Proved in: The logarithm function and its properties.
4. Monotonicity and sign of the logarithm. The function $\log_a$ is strictly increasing if $a > 1$ and strictly decreasing if $0 < a < 1$ , and $\log_a x > 0$ if and only if $(a - 1)(x - 1) > 0$ .
  Rests on: G (Monotonicity of the exponential), H (Logarithm as the inverse). Proved in: The logarithm function and its properties.
5. Irrational logarithms. If $a$ and $b$ are coprime natural numbers greater than $1$ , the number $\log_a b$ is irrational; in particular, so is $\log_2 3$ .
  Uses: 18. Monotonicity and sign of the logarithm. Rests on: F (Laws of exponents), H (Logarithm as the inverse), E (Euclid’s lemma). Proved in: The logarithm function and its properties.
6. Injectivity and equations. We have $a^u = a^v$ if and only if $u = v$ , and $\log_a u = \log_a v$ if and only if $u = v$ ; the equation $a^x = c$ has the unique solution $\log_a c$ if $c > 0$ , and none if $c \leq 0$ .
  Uses: 4. Exponential function. Rests on: H (Logarithm as the inverse). Proved in: Exponential and logarithmic equations and inequalities.
7. Monotonicity and inequalities. If $a > 1$ , we have $a^u < a^v$ if and only if $u < v$ , and $\log_a u < \log_a v$ if and only if $u < v$ ; if $0 < a < 1$ , the equivalences hold with the direction reversed.
  Uses: 18. Monotonicity and sign of the logarithm. Rests on: G (Monotonicity of the exponential). Proved in: Exponential and logarithmic equations and inequalities.
Segment 4 The models and the hyperbolic functions
- Characterize the model $P_0 e^{kt}$ by its constant rate over equal intervals and obtain the doubling time and the half-life.
- Linearize exponential data in the semi-log plane and study the logistic model.
- Define the hyperbolic cosine and sine from $e^x$ and $e^{-x}$ , and study their monotonicity.
- Prove the fundamental identity, parametrize the right branch of the hyperbola, and obtain the addition formulas and the inverse functions.
1. Constant rate. If $P(t) = P_0 e^{kt}$ with $P_0 > 0$ , then $P(t + h) = e^{kh} P(t)$ for all real numbers $t$ and $h$ ; at integer times it is the discrete model with $k = \ln(1 + r)$ , and the function increases if $k > 0$ and decreases if $k < 0$ .
  Rests on: F (Laws of exponents), G (Monotonicity of the exponential), H (Logarithm as the inverse). Proved in: Models of exponential growth and decay.
2. Converse of the constant rate. If $P$ is positive and monotonic and the quotient $\frac{P(t + h)}{P(t)}$ depends only on $h$ , then $P(t) = P_0 e^{kt}$ for some real $k$ : the equation $f(t + h) = f(t) f(h)$ and monotonicity determine the exponential.
  Uses: 2. Bracketing and monotonicity. Rests on: B (Density and Archimedean property), F (Laws of exponents), H (Logarithm as the inverse). Proved in: Models of exponential growth and decay.
3. Doubling and half-life. If $k > 0$ , the model $P_0 e^{kt}$ doubles over every interval of length $T = \frac{\ln 2}{k}$ , and only over those; if $k = -\lambda < 0$ , it is halved over every interval of length $\frac{\ln 2}{\lambda}$ .
  Uses: 22. Constant rate, 20. Injectivity and equations. Rests on: H (Logarithm as the inverse). Proved in: Models of exponential growth and decay.
4. Semi-log linearization. A point $(t, y)$ with $y > 0$ lies on the graph of $y = C e^{kt}$ if and only if $(t, \ln y)$ lies on the line $Y = \ln C + kt$ ; through two points with distinct abscissas passes a unique graph of that form.
  Uses: 16. Laws of logarithms. Rests on: H (Logarithm as the inverse). Proved in: Models of exponential growth and decay.
5. Logistic model. The function $P(t) = \frac{K}{1 + A e^{-rt}}$ , with $K$ , $A$ and $r$ positive, increases strictly between $0$ and $K$ , reaches $\frac{K}{2}$ at $t_0 = \frac{\ln A}{r}$ and its graph is symmetric with respect to that point.
  Uses: 4. Exponential function. Rests on: G (Monotonicity of the exponential), H (Logarithm as the inverse). Proved in: Models of exponential growth and decay.
6. Hyperbolic functions. With $\cosh x = \frac{e^x + e^{-x}}{2}$ and $\sinh x = \frac{e^x - e^{-x}}{2}$ , the hyperbolic cosine is even, the sine is odd, and $e^x = \cosh x + \sinh x$ , $e^{-x} = \cosh x - \sinh x$ .
  Uses: 4. Exponential function. Proved in: The hyperbolic functions.
7. Hyperbolic monotonicity. The hyperbolic sine and tangent are strictly increasing, with $-1 < \tanh x < 1$ ; the hyperbolic cosine decreases for $x \leq 0$ , increases for $x \geq 0$ and its minimum is $\cosh 0 = 1$ .
  Uses: 27. Hyperbolic functions. Rests on: F (Laws of exponents), G (Monotonicity of the exponential). Proved in: The hyperbolic functions.
8. Fundamental identity. For every real $x$ we have $\cosh^2 x - \sinh^2 x = 1$ .
  Uses: 27. Hyperbolic functions. Rests on: F (Laws of exponents). Proved in: The hyperbolic functions.
9. Parametrization of the hyperbola. The function $t \mapsto \left( \cosh t, \sinh t \right)$ is a bijection from $\mathbb{R}$ onto the right branch of $x^2 - y^2 = 1$ , and its inverse assigns to the point $(x, y)$ the number $\ln(x + y)$ .
  Uses: 20. Injectivity and equations, 27. Hyperbolic functions, 28. Hyperbolic monotonicity, 29. Fundamental identity. Rests on: H (Logarithm as the inverse). Proved in: The hyperbolic functions.
10. Addition formulas. We have $\cosh(a + b) = \cosh a \cosh b + \sinh a \sinh b$ and $\sinh(a + b) = \sinh a \cosh b + \cosh a \sinh b$ .
  Uses: 27. Hyperbolic functions. Rests on: F (Laws of exponents). Proved in: The hyperbolic functions.
11. Inverse hyperbolic functions. We have $\operatorname{arcsinh} x = \ln \left( x + \sqrt{x^2 + 1} \right)$ , $\operatorname{arccosh} x = \ln \left( x + \sqrt{x^2 - 1} \right)$ for $x \geq 1$ , and $\operatorname{arctanh} x = \frac{1}{2} \ln \frac{1 + x}{1 - x}$ for $-1 < x < 1$ .
  Uses: 28. Hyperbolic monotonicity. Rests on: H (Logarithm as the inverse). Proved in: The hyperbolic functions.

Towards the trigonometric functions. The chain closes on itself: the exponential produces the hyperbolic functions, and the logarithm inverts them. The following course will do with the circle $x^2 + y^2 = 1$ what this unit did with the hyperbola $x^2 - y^2 = 1$ .