Polynomials, Exponentials and Logarithms

1 Polynomials

ExrExercise 1 (Reading a polynomial)

Exercise 1 (Reading a polynomial) Let \(f(x) = 3x^4 - x + 7\).

1. Write \(f\) in the form \(a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0\), listing every coefficient \(a_4, a_3, a_2, a_1, a_0\).
2. What is the degree of \(f\), and what is its leading coefficient?

SolutionSolution 1

Solution 1.

1. \(f(x) = 3x^4 + 0 \cdot x^3 + 0 \cdot x^2 + (-1) \cdot x + 7\), so \(a_4 = 3\), \(a_3 = 0\), \(a_2 = 0\), \(a_1 = -1\), and \(a_0 = 7\). Powers of \(x\) that do not appear have coefficient \(0\).

2. The highest power with a nonzero coefficient is \(x^4\), so the degree is \(4\) and the leading coefficient is \(a_4 = 3\).

DefDefinition 1 (Constant function)

Definition 1 (Constant function) A function \(f : A \to B\) is constant if there is some \(c \in B\) with \(f(x) = c\) for all \(x \in A\).

ExmExample 1 (Constant and non-constant functions)

Example 1 (Constant and non-constant functions)  

• \(f(x) = 7\) on \(\mathbb{R}\) is constant, with \(c = 7\).
• \(f(x) = x^2\) on \(\mathbb{R}\) is not constant: \(f(0) = 0\) but \(f(1) = 1\).

DefDefinition 2 (Polynomial)

Definition 2 (Polynomial) A polynomial (in one variable \(x\)) [1] is a function \(f : \mathbb{R}\to \mathbb{R}\) that can be written as \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \] for some integer \(n \ge 0\) and constants \(a_0, a_1, \ldots, a_n \in \mathbb{R}\) with \(a_n \ne 0\). The terms of \(f\) are \(a_n x^n, \ldots, a_1 x, a_0\).

RemarkRemark 1 (Constant and zero polynomials)

Remark 1 (Constant and zero polynomials). A constant function \(f(x) = 7\) is a polynomial with \(n = 0\) and \(a_0 = 7\). The requirement \(a_n \ne 0\) means this definition covers nonzero polynomials only: the zero function \(f(x) = 0\) is excluded here, because it has no nonzero coefficient to serve as \(a_n\).

DefDefinition 3 (Coefficients of a polynomial)

Definition 3 (Coefficients of a polynomial) The constants \(a_0, a_1, \ldots, a_n\) of a polynomial \(f\) (Definition 2) are the coefficients of \(f\), with \(a_k\) the coefficient of \(x^k\).

DefDefinition 4 (Degree of a polynomial)

Definition 4 (Degree of a polynomial) Let \(f(x) = a_n x^n + \cdots + a_1 x + a_0\) be a polynomial with \(a_n \ne 0\). The integer \(n\) is the degree of \(f\).

ExmExample 2 (Degrees of some polynomials)

Example 2 (Degrees of some polynomials)  

• \(f(x) = 4x\) has degree \(1\).
• \(f(x) = 5 - 2x^2 + x^3\) has degree \(3\): the degree is the highest power with a nonzero coefficient, not the power in the first term written.
• A constant polynomial \(f(x) = 7\) has degree \(0\).

DefDefinition 5 (Leading coefficient)

Definition 5 (Leading coefficient) Let \(f(x) = a_n x^n + \cdots + a_1 x + a_0\) be a polynomial of degree \(n\). The constant \(a_n\) is the leading coefficient of \(f\).

ExmExample 3 (Leading coefficients)

Example 3 (Leading coefficients)  

• \(f(x) = 5 - 2x^2 + x^3\) has degree \(3\), so its leading coefficient is \(a_3 = 1\), not the \(5\) written first.
• \(f(x) = 3 - x^2\) has leading coefficient \(a_2 = -1\).

DefDefinition 6 (Quadratic polynomial)

Definition 6 (Quadratic polynomial) A polynomial of degree \(2\), \(f(x) = a_2 x^2 + a_1 x + a_0\) with \(a_2 \ne 0\), is a quadratic [2] polynomial.

DefDefinition 7 (Cubic polynomial)

Definition 7 (Cubic polynomial) A polynomial of degree \(3\) is a cubic polynomial.

ExmExample 4 (Quadratic and cubic polynomials)

Example 4 (Quadratic and cubic polynomials)  

• \[ \begin{aligned} f(x) &= (x - 2)^2 \\ &= x^2 - 4x + 4 \end{aligned} \]

  is quadratic, by Remark 5 in Convexity, Infimum and Sums with \(a = x\) and \(b = 2\).

• \(f(x) = x^3 - 3x\) is cubic.

• \(f(x) = 4x + 1\) is neither: it has degree \(1\).

DefDefinition 8 (Parabola)

Definition 8 (Parabola) A parabola [3] is the graph of a quadratic polynomial (Definition 6).

ExmExample 5 (The parabola \(y = x^2\))

Example 5 (The parabola \(y = x^2\)) The graph of \(f(x) = x^2\) is a parabola. It contains the points \((-1, 1)\), \((0, 0)\) and \((2, 4)\), and its lowest point is \((0, 0)\), since \(x^2 \ge 0\) for every \(x\).

DefDefinition 9 (Monomial (power product))

Definition 9 (Monomial (power product)) A monomial, also called a power product [4], in the variables \(x_1, \ldots, x_p\) is an expression

\[ x_1^{a_1} x_2^{a_2} \cdots x_p^{a_p} \tag{1}\]

that multiplies powers of those variables, whose exponents \(a_1, \ldots, a_p\) are nonnegative integers.

Each \(a_j = 0\) contributes the factor \(x_j^0 = 1\) (zero exponent).

ExmExample 6 (Which expressions are monomials)

Example 6 (Which expressions are monomials) Take the four variables \(x_1, x_2, x_3, x_4\).

• \(x_1^3 x_4\) is a monomial, with exponents \((a_1, a_2, a_3, a_4) = (3, 0, 0, 1)\).
• The constant \(1\) is the monomial whose exponents are all \(0\).
• \(x_1 / x_2 = x_1 x_2^{-1}\) is not a monomial, because the exponent of \(x_2\) is negative.
• \(\sqrt{x_3} = x_3^{1/2}\) is not a monomial, because its exponent is not an integer.
• \(x_1 + x_4\) is not a monomial: it adds two monomials rather than multiplying powers.

DefDefinition 10 (Degree of a monomial)

Definition 10 (Degree of a monomial) The degree of the monomial \(x_1^{a_1} x_2^{a_2} \cdots x_p^{a_p}\) is the sum of its exponents, \(a_1 + a_2 + \cdots + a_p\) [4].

ExmExample 7 (Degrees of some monomials)

Example 7 (Degrees of some monomials)  

• \(x_1^3 x_4\) has degree \(3 + 1 = 4\).
• \(x_2^2 x_3^5\) has degree \(2 + 5 = 7\).
• The constant monomial \(1\) has degree \(0\).
• With one variable, the monomial \(x^n\) has degree \(n\), which is the degree of the one-term polynomial \(f(x) = x^n\).

2 Affine and linear functions of one variable

DefDefinition 11 (Affine function of one variable)

Definition 11 (Affine function of one variable) A function \(f: \mathbb{R}\to \mathbb{R}\) is affine [5] if there are numbers \(m\) and \(b\) such that

\[f(x) = m x + b \quad \text{for all } x \in \mathbb{R}\]

DefDefinition 12 (Slope)

Definition 12 (Slope) In the formula \(f(x) = m x + b\) for an affine function \(f\) (Definition 11), \(m\) is called the slope of \(f\) [6].

DefDefinition 13 (Intercept)

Definition 13 (Intercept) In the formula \(f(x) = m x + b\) for an affine function \(f\) (Definition 11), \(b\) is called the intercept of \(f\).

DefDefinition 14 (Linear function of one variable)

Definition 14 (Linear function of one variable) A function \(f: \mathbb{R}\to \mathbb{R}\) is linear [7] if it is affine (Definition 11) with intercept \(0\); that is, if there is a number \(m\) such that

\[f(x) = m x \quad \text{for all } x \in \mathbb{R}\]

LemTheorem 1 (Intercept and slope of an affine function)

Theorem 1 (Intercept and slope of an affine function) Let \(f(x) = m x + b\) be an affine function (Definition 11).

1. The intercept is the value at zero: \(f(0) = b\).
2. The slope is the change in \(f\) per unit change in \(x\): for any \(x_1 \neq x_2\), \[\frac{f(x_2) - f(x_1)}{x_2 - x_1} = m\]
3. So \(f\) determines its slope and intercept: if also \(f(x) = m' x + b'\) for all \(x\), then \(m' = m\) and \(b' = b\).

ProofProof

Proof. Part 1.

\[ \begin{aligned} f(0) &= m \cdot 0 + b && \text{(}\href{#def-affine-function}{\text{Definition~11}}\text{)} \\&= 0 + b && \text{(any number times } 0 \text{ is } 0 \text{)} \\&= b + 0 && \text{(}\href{algebra-sums.qmd#thm-sum-symmetric}{\text{Theorem~5 in Convexity, Infimum and Sums}}\text{)} \\&= b && \text{(}\href{algebra-sums.qmd#thm-add-ident}{\text{Theorem~4 in Convexity, Infimum and Sums}}\text{)} \end{aligned} \]

Part 2. First the numerator:

\[ \begin{aligned} f(x_2) - f(x_1) &= (m x_2 + b) - (m x_1 + b) && \text{(}\href{#def-affine-function}{\text{Definition~11}}\text{)} \\&= m x_2 + b - m x_1 - b && \text{(subtracting a sum subtracts each term)} \\&= m x_2 - m x_1 + b - b && \text{(}\href{algebra-sums.qmd#thm-sum-symmetric}{\text{Theorem~5 in Convexity, Infimum and Sums}}\text{, swapping } b \text{ and } {-m x_1} \text{)} \\&= (m x_2 - m x_1) + (b - b) && \text{(}\href{algebra-sums.qmd#thm-sum-assoc}{\text{Theorem~6 in Convexity, Infimum and Sums}}\text{)} \\&= (m x_2 - m x_1) + 0 && (b - b = 0) \\&= m x_2 - m x_1 && \text{(}\href{algebra-sums.qmd#thm-add-ident}{\text{Theorem~4 in Convexity, Infimum and Sums}}\text{)} \\&= m x_2 + m (-x_1) && (-(m x_1) = m (-x_1) \text{, by }\href{algebra-basics.qmd#thm-negative-one}{\text{Theorem~6 in Equalities, Inequalities and Minimizers}}\text{ and }\href{algebra-sums.qmd#thm-prod-assoc}{\text{Theorem~9 in Convexity, Infimum and Sums}}\text{)} \\&= m (x_2 + (-x_1)) && \text{(}\href{algebra-sums.qmd#thm-mult-distr}{\text{Theorem~11 in Convexity, Infimum and Sums}}\text{, read right to left)} \\&= m (x_2 - x_1) && \text{(adding a negative is subtracting)} \end{aligned} \]

Then, writing \(d = x_2 - x_1\), which is not \(0\) because \(x_1 \neq x_2\),

\[ \begin{aligned} \frac{f(x_2) - f(x_1)}{x_2 - x_1} &= \frac{m d}{d} && \text{(the numerator above)} \\&= (m d) \cdot\frac{1}{d} && \text{(}\href{algebra-sums.qmd#thm-prod-div}{\text{Theorem~10 in Convexity, Infimum and Sums}}\text{)} \\&= m \cdot\mathopen{}\left(d \cdot\frac{1}{d}\right)\mathclose{} && \text{(}\href{algebra-sums.qmd#thm-prod-assoc}{\text{Theorem~9 in Convexity, Infimum and Sums}}\text{)} \\&= m \cdot\frac{d}{d} && \text{(}\href{algebra-sums.qmd#thm-prod-div}{\text{Theorem~10 in Convexity, Infimum and Sums}}\text{)} \\&= m \cdot 1 && \text{(a nonzero number divided by itself is } 1 \text{)} \\&= m && \text{(}\href{algebra-sums.qmd#thm-mult-one}{\text{Theorem~7 in Convexity, Infimum and Sums}}\text{)} \end{aligned} \]

Part 3. Part 1, applied to each formula, gives

\[ \begin{aligned} b &= f(0) \\ &= b'. \end{aligned} \]

Part 2 with \(x_1 = 0\) and \(x_2 = 1\), applied to each formula, gives

\[ \begin{aligned} m &= f(1) - f(0) \\ &= m'. \end{aligned} \]

ExmExample 8 (An affine function, and a linear one)

Example 8 (An affine function, and a linear one) \(f(x) = 2x + 3\) is affine with slope \(2\) and intercept \(3\). By Theorem 1, \(f(0) = 3\), and each unit step in \(x\) raises \(f\) by \(2\): for example, \(f(1) = 5\) and \(f(2) = 7\). Its intercept is not \(0\), so it is not linear.

\(g(x) = -\tfrac{1}{2} x\) is affine with intercept \(0\), so it is linear: \(g(0) = 0\), and each unit step in \(x\) lowers \(g\) by \(\tfrac{1}{2}\).

RemarkRemark 2 (Elementary algebra calls \(m x + b\) “linear”)

Remark 2 (Elementary algebra calls \(m x + b\) “linear”). Elementary algebra usually calls \(f(x) = m x + b\) a linear function, because its graph is a straight line. This site calls that an affine function and keeps “linear” for the case \(b = 0\), because that is the sense used in linear algebra, where a linear map must send \(0\) to \(0\) (theorem).

3 Limits of sequences

DefDefinition 15 (Limit of a sequence)

Definition 15 (Limit of a sequence) A real number \(L\) is the limit of a sequence \((a_n)\) of real numbers [8], written \(\lim_{n \to \infty} a_n = L\) or \(a_n \to L\), if for every \(\varepsilon> 0\) there is a natural number \(N\) such that

\[\mathopen{}\left|a_n - L\right|\mathclose{} < \varepsilon\quad \text{for every } n \ge N.\]

ExmExample 9 (The limit of a sequence)

Example 9 (The limit of a sequence) \(a_n = \frac{1}{n}\) has limit \(0\). Given \(\varepsilon> 0\), take \(N\) to be any natural number larger than \(\frac{1}{\varepsilon}\). For every \(n \ge N\), \(\mathopen{}\left|\frac{1}{n} - 0\right|\mathclose{} = \frac{1}{n} \le \frac{1}{N} < \varepsilon\). For example, with \(\varepsilon= 0.01\), take \(N = 101\): every \(n \ge 101\) has \(\frac{1}{n} \le \frac{1}{101} < 0.01\).

DefDefinition 16 (Convergent sequence (converges))

Definition 16 (Convergent sequence (converges)) A sequence \((a_n)\) of real numbers converges to a real number \(L\) if \(L\) is its limit (Definition 15). A sequence that converges to some real number is convergent.

ExmExample 10 (A convergent sequence)

Example 10 (A convergent sequence) \(a_n = \frac{1}{n}\) converges to \(0\), because \(0\) is its limit (Example 9), so \(a_n = \frac{1}{n}\) is convergent.

DefDefinition 17 (Divergent sequence (diverges))

Definition 17 (Divergent sequence (diverges)) A sequence that does not converge (Definition 16) to any real number diverges, and a sequence that diverges is divergent.

ExmExample 11 (A divergent sequence)

Example 11 (A divergent sequence) \(c_n = n\) diverges. For any real number \(L\) and \(\varepsilon= 1\), every \(n > L + 1\) has \(\mathopen{}\left|n - L\right|\mathclose{} > 1\), so no \(N\) works. For example, with \(L = 5\), every \(n \ge 7\) has \(\mathopen{}\left|n - 5\right|\mathclose{} \ge 2\). It diverges to \(\infty\): for every real number \(M\), every \(n \ge N\) has \(c_n = n > M\) when \(N\) is a natural number larger than \(M\).

DefDefinition 18 (Divergence of a sequence to infinity)

Definition 18 (Divergence of a sequence to infinity) A sequence \((a_n)\) of real numbers diverges to \(\infty\), written \(\lim_{n \to \infty} a_n = \infty\), if for every real number \(M\) there is a natural number \(N\) such that \(a_n > M\) for every \(n \ge N\).

4 Series

DefDefinition 19 (Non-decreasing sequence)

Definition 19 (Non-decreasing sequence) A sequence \(a_1, a_2, \ldots\) of real numbers or of extended non-negative reals is non-decreasing if each term is at least as large as the one before it:

\[a_n \le a_{n+1} \quad \text{for every } n \in \mathbb{N}\]

ExmExample 12 (Non-decreasing and not)

Example 12 (Non-decreasing and not)  

• \(1, 2, 2, 3, 3, 3, \ldots\) is non-decreasing: a term may equal the one before it.

• \(1, 2, \infty, \infty, \ldots\) is non-decreasing, because \(\infty\) is greater than every real number.

• \(1, \frac{1}{2}, \frac{1}{3}, \ldots\) is not non-decreasing, because

  \[ \begin{aligned} a_2 &= \frac{1}{2} \\ &< 1 \\ &= a_1. \end{aligned} \]

DefDefinition 20 (Partial sum)

Definition 20 (Partial sum) Let \(a_1, a_2, \ldots\) be a sequence of real numbers or of extended non-negative reals. For each \(n \in \mathbb{N}\), its \(n\)th partial sum [9] is the sum of its first \(n\) terms:

\[s_n \stackrel{\text{def}}{=}\sum_{i=1}^na_i = a_1 + \cdots + a_n\]

ExmExample 13 (Partial sums of halves)

Example 13 (Partial sums of halves) For \(a_i = 1/2^i\), the sequence \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \ldots\), the first three partial sums are \(s_1 = \frac{1}{2}\),

\[ \begin{aligned} s_2 &= \frac{1}{2} + \frac{1}{4} \\ &= \frac{3}{4}, \end{aligned} \]

and

\[ \begin{aligned} s_3 &= \frac{3}{4} + \frac{1}{8} \\ &= \frac{7}{8}. \end{aligned} \]

In general, \(s_n = 1 - 1/2^n\).

DefDefinition 21 (Infinite sum, series)

Definition 21 (Infinite sum, series) Let \(a_1, a_2, \ldots\) be a sequence in \([0, \infty]\), with partial sums \(s_n\) (Definition 20). The infinite sum (also called a series) [9] \(\sum_{i=1}^{\infty} a_i\) is the limit of the partial sums:

\[\sum_{i=1}^{\infty} a_i \stackrel{\text{def}}{=}\lim_{n \to \infty} s_n\]

Here the limit is a real number when the partial sums are real numbers that converge to it, and it is \(\infty\) when the partial sums diverge to \(\infty\) or some partial sum is \(\infty\). Sums with other index ranges, such as \(\sum_{k=0}^{\infty} a_k\), are defined the same way.

ExmExample 14 (A finite and an infinite sum)

Example 14 (A finite and an infinite sum)  

• For \(a_i = 1/2^i\), the partial sums \(s_n = 1 - 1/2^n\) (Example 13) converge to \(1\), because \(1/2^n \to 0\). So \(\sum_{i=1}^{\infty} 1/2^i = 1\).
• For \(a_i = 1\), the partial sums are \(s_n = n\), which diverge to \(\infty\). So \(\sum_{i=1}^{\infty} 1 = \infty\).

LemLemma 1 (Sums of non-negative terms)

Lemma 1 (Sums of non-negative terms) Let \(a_1, a_2, \ldots\) be values in \([0, \infty]\), with partial sums \(s_n\) (Definition 20), where \(x + \infty = \infty\) for every \(x\) in \([0, \infty]\). Then the partial sums are non-decreasing, \(s_1 \le s_2 \le \cdots\), so their limit, the infinite sum \(\sum_{i=1}^{\infty} a_i\) (Definition 21), always exists, and it is either a finite number or \(\infty\).

ProofProof

Proof. For each \(n\):

\[ \begin{aligned} s_{n+1} &= s_n + a_{n+1} && \text{(definition of } s_{n+1} \text{)} \\ &\ge s_n && \text{(} a_{n+1} \ge 0 \text{)} \end{aligned} \]

If some partial sum \(s_N\) is \(\infty\), then \(s_n = \infty\) for every \(n \ge N\), so the limit is \(\infty\). Otherwise, the partial sums form a non-decreasing sequence of real numbers. If that sequence is bounded above, it converges to a finite number [10]. If it is not bounded above, then for every number \(M\) some \(s_N\) exceeds \(M\), and so does every later \(s_n \ge s_N\); the limit is \(\infty\).

5 Exponentials and Logarithms

DefDefinition 22 (Exponential function)

Definition 22 (Exponential function) The exponential function [11] \(\operatorname{exp}: \mathbb{R}\to (0, \infty)\) is

\[{\operatorname{exp}\mathopen{}\left\{x\right\}\mathclose{}} \stackrel{\text{def}}{=}\lim_{n \to \infty} \mathopen{}\left(1 + \frac{x}{n}\right)\mathclose{}^n,\]

the limit of the sequence \(\mathopen{}\left(1 + \frac{x}{1}\right)\mathclose{}^1, \mathopen{}\left(1 + \frac{x}{2}\right)\mathclose{}^2, \mathopen{}\left(1 + \frac{x}{3}\right)\mathclose{}^3, \ldots\). That limit exists for every real number \(x\), and it is positive.

ExmExample 15 (Approximating \({\operatorname{exp}\mathopen{}\left\{1\right\}\mathclose{}}\))

Example 15 (Approximating \({\operatorname{exp}\mathopen{}\left\{1\right\}\mathclose{}}\)) For \(x = 1\), the terms \(\mathopen{}\left(1 + \frac{1}{n}\right)\mathclose{}^n\) are:

\(n\)	\(\mathopen{}\left(1 + \frac{1}{n}\right)\mathclose{}^n\)
\(1\)	\(2\)
\(10\)	\(2.59374\ldots\)
\(100\)	\(2.70481\ldots\)
\(1000\)	\(2.71692\ldots\)

They approach \({\operatorname{exp}\mathopen{}\left\{1\right\}\mathclose{}} = 2.71828\ldots\). For \(x = 0\), every term is \(\mathopen{}\left(1 + 0\right)\mathclose{}^n = 1\), so \({\operatorname{exp}\mathopen{}\left\{0\right\}\mathclose{}} = 1\).

DefDefinition 23 (Euler’s number)

Definition 23 (Euler’s number) Euler’s number [12] is

\[e \stackrel{\text{def}}{=}{\operatorname{exp}\mathopen{}\left\{1\right\}\mathclose{}} = 2.71828\ldots\]

ExmExample 16 (Euler’s number is irrational)

Example 16 (Euler’s number is irrational) \(e = 2.71828\ldots\) is an irrational number, so no fraction equals it exactly. The fraction \(\frac{19}{7} = 2.714\ldots\) is close, but \(\frac{19}{7} \ne e\) [12].

DefDefinition 24 (Natural logarithm)

Definition 24 (Natural logarithm) For a real number \(a > 0\), the natural logarithm of \(a\), written \({\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}}\) [13], is the real number \(y\) with \({\operatorname{exp}\mathopen{}\left\{y\right\}\mathclose{}} = a\). The exponential function takes each positive value at exactly one input, so there is exactly one such \(y\), and \(\log : (0, \infty) \to \mathbb{R}\) is the inverse function of \(\operatorname{exp}\) (Definition 22).

RemarkRemark 3 (Other notations for the natural logarithm)

Remark 3 (Other notations for the natural logarithm). In these notes, \({\operatorname{log}\mathopen{}\left\{x\right\}\mathclose{}}\) is always the natural logarithm, the logarithm with base \(e\) (Definition 23) [14]. Some sources write \(\ln x\) for the natural logarithm and reserve \(\log x\) for the logarithm with base 10.

ExmExample 17 (Natural logarithms)

Example 17 (Natural logarithms)  

• \({\operatorname{log}\mathopen{}\left\{1\right\}\mathclose{}} = 0\), because \({\operatorname{exp}\mathopen{}\left\{0\right\}\mathclose{}} = 1\) (Example 15).
• \({\operatorname{log}\mathopen{}\left\{e\right\}\mathclose{}} = 1\), because \({\operatorname{exp}\mathopen{}\left\{1\right\}\mathclose{}} = e\) (Definition 23).
• \({\operatorname{log}\mathopen{}\left\{0\right\}\mathclose{}}\) and \({\operatorname{log}\mathopen{}\left\{-2\right\}\mathclose{}}\) are not defined, because \({\operatorname{exp}\mathopen{}\left\{y\right\}\mathclose{}} > 0\) for every real \(y\).

DefDefinition 25 (Power with a real exponent)

Definition 25 (Power with a real exponent) For a real number \(a > 0\) and a real number \(b\),

\[a^b \stackrel{\text{def}}{=}{\operatorname{exp}\mathopen{}\left\{b \cdot{\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}}\right\}\mathclose{}}.\]

When \(b\) is an integer, this agrees with Definition 5 in Equalities, Inequalities and Minimizers and Definition 9 in Equalities, Inequalities and Minimizers.

ExmExample 18 (Real powers, and \(e^x\))

Example 18 (Real powers, and \(e^x\))  

• \(2^3 = {\operatorname{exp}\mathopen{}\left\{3 \cdot{\operatorname{log}\mathopen{}\left\{2\right\}\mathclose{}}\right\}\mathclose{}} \approx {\operatorname{exp}\mathopen{}\left\{3 \cdot 0.69315\right\}\mathclose{}} \approx {\operatorname{exp}\mathopen{}\left\{2.07944\right\}\mathclose{}} \approx 8\), which agrees with

  \[ \begin{aligned} 2^3 &= 2 \cdot 2 \cdot 2 \\ &= 8 \end{aligned} \]

  from Definition 5 in Equalities, Inequalities and Minimizers.

• \(2^{1/2} = {\operatorname{exp}\mathopen{}\left\{\frac{1}{2} \cdot{\operatorname{log}\mathopen{}\left\{2\right\}\mathclose{}}\right\}\mathclose{}} \approx {\operatorname{exp}\mathopen{}\left\{0.34657\right\}\mathclose{}} \approx 1.41421\), which is \(\sqrt{2}\) (Definition 10 in Equalities, Inequalities and Minimizers).

• For every real \(x\), with base \(e\) (Definition 23):

  \[ \begin{aligned} e^x &= {\operatorname{exp}\mathopen{}\left\{x \cdot{\operatorname{log}\mathopen{}\left\{e\right\}\mathclose{}}\right\}\mathclose{}} && \text{(}\href{#def-real-power}{\text{Definition~25}}\text{, with } a = e \text{ and } b = x \text{)} \\ &= {\operatorname{exp}\mathopen{}\left\{x \cdot 1\right\}\mathclose{}} && \text{(}{\operatorname{log}\mathopen{}\left\{e\right\}\mathclose{}} = 1 \text{, by }\href{#exm-natural-log}{\text{Example~17}}\text{)} \\ &= {\operatorname{exp}\mathopen{}\left\{x\right\}\mathclose{}} && \text{(}\href{algebra-sums.qmd#thm-mult-one}{\text{Theorem~7 in Convexity, Infimum and Sums}}\text{)} \end{aligned} \]

  So \(e^x\) and \({\operatorname{exp}\mathopen{}\left\{x\right\}\mathclose{}}\) are two names for the same number.

DefDefinition 26 (Real powers of zero and of negative numbers)

Definition 26 (Real powers of zero and of negative numbers) Definition 5 in Equalities, Inequalities and Minimizers, Definition 9 in Equalities, Inequalities and Minimizers and Definition 25 leave some powers \(b^c\) with \(b \le 0\) unassigned. We complete them as follows.

• For a real number \(c > 0\), \(0^c \stackrel{\text{def}}{=}0\). For a natural number \(c\), this agrees with Definition 5 in Equalities, Inequalities and Minimizers.
• For \(b < 0\), \(b^c\) is defined only when \(c\) is an integer, by Definition 5 in Equalities, Inequalities and Minimizers and Definition 9 in Equalities, Inequalities and Minimizers; for \(b < 0\) and a non-integer \(c\), \(b^c\) is undefined.

ExmExample 19 (Powers of zero and of negative numbers)

Example 19 (Powers of zero and of negative numbers)  

• \(0^{1/2} = 0\), which agrees with \(\sqrt{0} = 0\) (Definition 10 in Equalities, Inequalities and Minimizers).
• \((-8)^2 = 64\) and \((-8)^{-1} = -\tfrac{1}{8}\) are defined, since the exponents are integers.
• \((-8)^{1/3}\) is undefined, even though \((-2)^3 = -8\). R follows the same convention: (-8)^(1/3) is NaN.

LemTheorem 2 (\(\operatorname{exp}\) and \(\operatorname{log}\) are mutual inverses)

Theorem 2 (\(\operatorname{exp}\) and \(\operatorname{log}\) are mutual inverses)  

1. For every \(a > 0\): \({\operatorname{exp}\mathopen{}\left\{{\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}}\right\}\mathclose{}} = a\).
2. For every \(a \in \mathbb{R}\): \({\operatorname{log}\mathopen{}\left\{{\operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{}}\right\}\mathclose{}} = a\).

LemTheorem 3 (Logarithm of a product)

Theorem 3 (Logarithm of a product) If \(a > 0\) and \(b > 0\), then

\[ {\operatorname{log}\mathopen{}\left\{a \cdot b\right\}\mathclose{}} = {\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} + {\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}} \]

CorCorollary 1 (Logarithm of a quotient)

Corollary 1 (Logarithm of a quotient) If \(a > 0\) and \(b > 0\), then

\[{\operatorname{log}\mathopen{}\left\{\frac{a}{b}\right\}\mathclose{}} = {\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}}\]

ProofProof

Proof. Since \(a > 0\) and \(b > 0\), the quotient \(\frac{a}{b}\) is positive, so Theorem 3 applies to the product \(\frac{a}{b} \cdot b\):

\[ \begin{aligned} {\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} &= {\operatorname{log}\mathopen{}\left\{\frac{a}{b} \cdot b\right\}\mathclose{}} && \text{(} a = \tfrac{a}{b} \cdot b \text{)} \\ &= {\operatorname{log}\mathopen{}\left\{\frac{a}{b}\right\}\mathclose{}} + {\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}} && \text{(logarithm of a product)} \end{aligned} \]

The second step applies Theorem 3. Subtracting \({\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}}\) from both sides gives \({\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} - {\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}} = {\operatorname{log}\mathopen{}\left\{\frac{a}{b}\right\}\mathclose{}}\).

LemTheorem 4 (Logarithm of a power)

Theorem 4 (Logarithm of a power) If \(a > 0\) and \(b \in \mathbb{R}\), then

\[ {\operatorname{log}\mathopen{}\left\{a^b\right\}\mathclose{}} = b \cdot{\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} \]

LemTheorem 5 (Exponential of a sum)

Theorem 5 (Exponential of a sum) The exponential of a sum is equal to the product of the exponentials of its terms:

\[{\operatorname{exp}\mathopen{}\left\{a+b\right\}\mathclose{}} = {\operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{}} \cdot{\operatorname{exp}\mathopen{}\left\{b\right\}\mathclose{}}\]

ExmExample 20 (Exponential of a sum)

Example 20 (Exponential of a sum) With \(a = 2\) and \(b = 3\), \({\operatorname{exp}\mathopen{}\left\{2 + 3\right\}\mathclose{}} = {\operatorname{exp}\mathopen{}\left\{5\right\}\mathclose{}} \approx 148.41\), and \({\operatorname{exp}\mathopen{}\left\{2\right\}\mathclose{}} \cdot{\operatorname{exp}\mathopen{}\left\{3\right\}\mathclose{}} \approx 7.3891 \cdot 20.0855 \approx 148.41\).

CorCorollary 2 (Exponential of a difference)

Corollary 2 (Exponential of a difference) The exponential of a difference is the exponential of the first term divided by the exponential of the second term:

\[{\operatorname{exp}\mathopen{}\left\{a-b\right\}\mathclose{}} = \frac{{\operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{}}}{{\operatorname{exp}\mathopen{}\left\{b\right\}\mathclose{}}}\]

ExmExample 21 (Exponential of a difference)

Example 21 (Exponential of a difference) With \(a = 5\) and \(b = 2\), \({\operatorname{exp}\mathopen{}\left\{5 - 2\right\}\mathclose{}} = {\operatorname{exp}\mathopen{}\left\{3\right\}\mathclose{}} \approx 20.09\), and \(\frac{{\operatorname{exp}\mathopen{}\left\{5\right\}\mathclose{}}}{{\operatorname{exp}\mathopen{}\left\{2\right\}\mathclose{}}} \approx \frac{148.4132}{7.3891} \approx 20.09\).

LemTheorem 6 (Powers of 1 and first powers)

Theorem 6 (Powers of 1 and first powers) For every \(b \in \mathbb{R}\),

\[1^b = 1,\]

and for every \(a \in \mathbb{R}\),

\[a^1 = a.\]

LemTheorem 7 (Power of a sum)

Theorem 7 (Power of a sum) If \(a > 0\) and \(b, c \in \mathbb{R}\), then

\[a^{b+c} = a^b \cdot a^c\]

ExmExample 22 (Power of a sum)

Example 22 (Power of a sum) With \(a = 2\), \(b = 3\), and \(c = 4\),

\[ \begin{aligned} 2^{3+4} &= 2^7 \\ &= 128, \end{aligned} \]

and

\[ \begin{aligned} 2^3 \cdot 2^4 &= 8 \cdot 16 \\ &= 128. \end{aligned} \]

LemTheorem 8 (Power of a product)

Theorem 8 (Power of a product) If \(a, b > 0\) and \(c \in \mathbb{R}\), then

\[(ab)^c = a^c \cdot b^c\]

When \(c\) is a positive integer, the same identity holds for all \(a, b \in \mathbb{R}\), because both sides are products of \(c\) copies of \(a\) and \(c\) copies of \(b\), which can be regrouped by Theorem 8 in Convexity, Infimum and Sums and Theorem 9 in Convexity, Infimum and Sums.

ExmExample 23 (Power of a product)

Example 23 (Power of a product) With \(a = 2\), \(b = 3\), and \(c = 2\),

\[ \begin{aligned} (2 \cdot 3)^2 &= 6^2 \\ &= 36, \end{aligned} \]

and

\[ \begin{aligned} 2^2 \cdot 3^2 &= 4 \cdot 9 \\ &= 36. \end{aligned} \]

LemTheorem 9 (Power of a power)

Theorem 9 (Power of a power) If \(a > 0\) and \(b, c \in \mathbb{R}\), then

\[ \begin{aligned} a^{bc} &= \mathopen{}\left(a^b\right)\mathclose{}^c \\ &= \mathopen{}\left(a^c\right)\mathclose{}^b \end{aligned} \]

ExmExample 24 (A negative base)

Example 24 (A negative base) With \(a = -1\), \(b = 2\), and \(c = \frac{1}{2}\):

\[ \begin{aligned} a^{bc} &= (-1)^{2 \cdot\frac{1}{2}} \\ &= (-1)^{1} \\ &= -1 \end{aligned} \]

but

\[ \begin{aligned} \mathopen{}\left(a^b\right)\mathclose{}^c &= \mathopen{}\left((-1)^2\right)\mathclose{}^{\frac{1}{2}} \\ &= 1^{\frac{1}{2}} \\ &= 1 \end{aligned} \]

So \(a^{bc} \neq \mathopen{}\left(a^b\right)\mathclose{}^c\) here, which is why Theorem 9 requires \(a > 0\). The third expression, \(\mathopen{}\left(a^c\right)\mathclose{}^b = \mathopen{}\left((-1)^{\frac{1}{2}}\right)\mathclose{}^2\), is not even a real number.

CorCorollary 3 (Natural exponential of a product)

Corollary 3 (Natural exponential of a product) \[ \begin{aligned} {\operatorname{exp}\mathopen{}\left\{ab\right\}\mathclose{}} &= ({\operatorname{exp}\mathopen{}\left\{a\right\}\mathclose{}})^b \\ &= ({\operatorname{exp}\mathopen{}\left\{b\right\}\mathclose{}})^a \end{aligned} \]

RemarkRemark 4 (Tarski’s high school identities)

Remark 4 (Tarski’s high school identities). Restricted to positive integers, the following results are Tarski’s eleven “high school” identities [15]:

• sums are symmetric and associative (Theorem 5 in Convexity, Infimum and Sums, Theorem 6 in Convexity, Infimum and Sums);
• multiplying by 1 changes nothing, products are symmetric and associative, and multiplication is distributive (Theorem 7 in Convexity, Infimum and Sums, Theorem 8 in Convexity, Infimum and Sums, Theorem 9 in Convexity, Infimum and Sums, Theorem 11 in Convexity, Infimum and Sums);
• \(1^b = 1\), \(a^1 = a\), and the power of a sum, of a product, and of a power (Theorem 6, Theorem 7, Theorem 8, Theorem 9).

RemarkRemark 5 (Tarski’s identities are not complete)

Remark 5 (Tarski’s identities are not complete). Tarski asked whether every identity in \(+\), \(\times\), exponentiation and 1 that is true for all positive integers can be derived from the eleven identities in Remark 4. It cannot: Wilkie found an identity that is true for all positive integers but does not follow from them. So this list is a useful core, not a complete rulebook.

DefDefinition 27 (Contour line)

Definition 27 (Contour line) For a function \(g\) of two real variables \(b\) and \(c\) and a number \(k\), the contour line [16] of \(g\) at height \(k\) is the set of points where \(g\) equals \(k\):

\[\mathopen{}\left\{(b, c) : g(b, c) = k\right\}\mathclose{}\]

ExmExample 25 (Contour lines of a bowl)

Example 25 (Contour lines of a bowl) For \(g(b, c) = b^2 + c^2\), the contour line at height \(k = 4\) is \(\mathopen{}\left\{(b, c) : b^2 + c^2 = 4\right\}\mathclose{}\), the circle of radius \(2\) around \((0, 0)\): for example,

\[ \begin{aligned} g(2, 0) &= 4 + 0 \\ &= 4 \end{aligned} \]

and

\[ \begin{aligned} g(0, -2) &= 0 + 4 \\ &= 4. \end{aligned} \]

The contour line at height \(k = -1\) is empty, because \(b^2 + c^2 \ge 0\) for all \(b\) and \(c\).

library(ggplot2)

# The bowl g(b, c) = b^2 + c^2 on a grid, and the two points of the example.
bowl <- expand.grid(
  b = seq(-3.5, 3.5, length.out = 200),
  c = seq(-3.5, 3.5, length.out = 200)
)
bowl$g <- bowl$b^2 + bowl$c^2
on_circle <- data.frame(b = c(2, 0), c = c(0, -2))

contour_plot <- ggplot(bowl) +
  aes(b, c) +
  geom_contour(
    aes(z = g, colour = factor(after_stat(level))),
    breaks = c(1, 4, 9)
  ) +
  geom_point(data = on_circle, size = 2.5) +
  scale_colour_viridis_d(name = "height k", end = 0.85) +
  coord_fixed(expand = FALSE) +
  labs(x = "first coordinate, b", y = "second coordinate, c") +
  theme(legend.position = "bottom")

if (knitr::is_html_output()) {
  htmltools::div(
    role = "img",
    `aria-label` = fig_alt_contour_bowl,
    plotly::ggplotly(contour_plot) |>
      plotly::layout(
        legend = list(orientation = "h", x = 0.5, xanchor = "center", y = -0.25)
      ) |>
      plotly::config(displaylogo = FALSE, responsive = TRUE)
  )
} else {
  contour_plot
}

Figure 1: The contour lines of \(g(b, c) = b^2 + c^2\) at heights \(1\), \(4\) and \(9\). The dots mark the points \((2, 0)\) and \((0, -2)\) of the example on the circle at height \(4\).

RemarkRemark 6 (Contour lines on a plot of a surface)

Remark 6 (Contour lines on a plot of a surface). On a plot of the surface \(z = g(b, c)\), the contour line is the curve along which the surface has height \(k\).

ExrExercise 2 (Exponentiation versus multiplication)

Exercise 2 (Exponentiation versus multiplication) For \(b,c \in \mathbb{R}\), when does \(b^c = bc\)?

SolutionSolution 2

Solution 2. We only count a pair \((b, c)\) when \(b^c\) is defined, so for \(b < 0\) we only consider integer \(c\) (Definition 26). With that convention, \(bc = b^c\) in each of the following cases:

1. \(c = 1\), for every \(b\).

2. \(b = 0\) and \(c > 0\), since then

   \[ \begin{aligned} b^c &= 0 \\ &= bc. \end{aligned} \]

   (\(b = 0\) and \(c = 0\) fails, since \(0^0 = 1 \neq 0\).)

3. \(b > 0\), \(c > 0\), \(c \neq 1\), and \(b = {\operatorname{exp}\mathopen{}\left\{\frac{{\operatorname{log}\mathopen{}\left\{c\right\}\mathclose{}}}{c-1}\right\}\mathclose{}}\). For \(b > 0\), dividing both sides of \(b^c = bc\) by \(b\) gives \(b^{c-1} = c\), which needs \(c > 0\) because \(b^{c-1} > 0\); taking logarithms then gives \((c-1){\operatorname{log}\mathopen{}\left\{b\right\}\mathclose{}} = {\operatorname{log}\mathopen{}\left\{c\right\}\mathclose{}}\). For example, \(c = 2\) gives \(b = 2\), and indeed

   \[ \begin{aligned} 2^2 &= 4 \\ &= 2 \cdot 2. \end{aligned} \]

4. \(b < 0\) and \(c\) is an odd integer with \(c \ge 3\), with \(b = -c^{1/(c-1)}\); for example, \(b = -\sqrt{3}\) and \(c = 3\) give

   \[ \begin{aligned} b^c &= -3\sqrt{3} \\ &= bc. \end{aligned} \]

5. \(b < 0\) and \(c\) is an even integer with \(c \le -2\), with \(b = -(-c)^{1/(c-1)}\); for example, \(b = -2^{-1/3}\) and \(c = -2\) give

   \[ \begin{aligned} b^c &= 2^{2/3} \\ &= bc. \end{aligned} \]

For \(b < 0\), cases 4 and 5 come from \(b^{c-1} = c\) as well: when \(c - 1\) is even, \(b^{c-1} > 0\), so \(c\) must be positive; when \(c - 1\) is odd, \(b^{c-1} < 0\), so \(c\) must be negative.

See the red contours in Figure 3 for a visualization of the \(b \ge 0\) cases.

mult_f <- function(b, c) b * c
pow_f <- function(b, c) b^c
values_b <- seq(0, 5, by = .01)
values_c <- seq(-.5, 3, by = .01)

mult_mat <- outer(values_b, values_c, mult_f)
pow_mat <- outer(values_b, values_c, pow_f)
pow_mat[is.infinite(pow_mat)] <- NA

opacity <- .3
z_min <- min(mult_mat, pow_mat, na.rm = TRUE)
z_max <- 5
plotly::plot_ly(
  x = ~values_b,
  y = ~values_c
) |>
  plotly::add_surface(
    z = ~ t(mult_mat),
    contours = list(
      z = list(
        show = TRUE,
        start = -1,
        end = 1,
        size = .1
      )
    ),
    name = "b*c",
    showscale = FALSE,
    opacity = opacity,
    colorscale = list(c(0, 1), c("green", "green"))
  ) |>
  plotly::add_surface(
    opacity = opacity,
    colorscale = list(c(0, 1), c("red", "red")),
    z = ~ t(pow_mat),
    contours = list(
      z = list(
        show = TRUE,
        start = z_min,
        end = z_max,
        size = .2
      )
    ),
    showscale = FALSE,
    name = "b^c"
  ) |>
  plotly::layout(
    scene = list(
      xaxis = list(
        # type = "log",
        title = "First input, b"
      ),
      yaxis = list(
        # type = "log",
        title = "Second input, c"
      ),
      zaxis = list(
        # type = "log",
        range = c(z_min, z_max),
        title = "outcome"
      ),
      camera = list(eye = list(x = -1.25, y = -1.25, z = 0.5)),
      aspectratio = list(x = .9, y = .8, z = 0.7)
    )
  )

Figure 2: Graph of \(b*c\) and \(b^c\)

pow_minus_mult_f <- function(b, c) pow_f(b, c) - mult_f(b, c)

mat1 <- outer(values_b, values_c, pow_minus_mult_f)
mat1[is.infinite(mat1)] <- NA

opacity <- .3
plotly::plot_ly(
  x = ~values_b,
  y = ~values_c
) |>
  plotly::add_surface(
    z = ~ t(mat1),
    contours = list(
      z = list(
        show = TRUE,
        start = 0,
        end = 1,
        size = 1,
        color = "red"
      )
    ),
    name = "b^c - b*c",
    showscale = TRUE,
    opacity = opacity
  ) |>
  plotly::layout(
    scene = list(
      xaxis = list(
        # type = "log",
        title = "First input, b"
      ),
      yaxis = list(
        # type = "log",
        title = "Second input, c"
      ),
      zaxis = list(
        title = "outcome"
      ),
      camera = list(eye = list(x = -1.25, y = -1.25, z = 0.5)),
      aspectratio = list(x = .9, y = .8, z = 0.7)
    )
  )

Figure 3: Graph of \(b^c - b*c\). The red contour lines are at heights \(0\) and \(1\); the ones at height \(0\) show where \(b^c = b*c\).

LemTheorem 10 (When exponentiation equals multiplication)

Theorem 10 (When exponentiation equals multiplication) Let \(b, c \in \mathbb{R}\), where \(b^c\) is defined (for \(b < 0\), only integer \(c\) is allowed; Definition 26). Then \(b^c = bc\) if and only if one of the following holds:

1. \(c = 1\);
2. \(b = 0\) and \(c > 0\);
3. \(b > 0\), \(c > 0\), \(c \neq 1\), and \(b = {\operatorname{exp}\mathopen{}\left\{\frac{{\operatorname{log}\mathopen{}\left\{c\right\}\mathclose{}}}{c-1}\right\}\mathclose{}}\);
4. \(b < 0\), \(c\) is an odd integer with \(c \ge 3\), and \(b = -c^{1/(c-1)}\);
5. \(b < 0\), \(c\) is an even integer with \(c \le -2\), and \(b = -(-c)^{1/(c-1)}\).

ProofProof

Proof. This is the solution to Exercise 2, worked in Solution 2.

ExrExercise 3 (Repeated exponentiation)

Exercise 3 (Repeated exponentiation) For \(a \ge 0,~b,c \in \mathbb{R}\), when does \((a^b)^c = a^{(b^c)}\)?

SolutionSolution 3

Solution 3. Short answer: rarely (that’s all you need to know for this course).

Long answer:

Split on whether \(a > 0\) or \(a = 0\), because the logarithm we use for \(a > 0\) is undefined at \(a = 0\).

Case \(a > 0\). By Theorem 9, \((a^b)^c = a^{bc}\), so the question becomes when \(a^{bc} = a^{(b^c)}\) (for pairs \((b, c)\) where \(b^c\) is defined). Because \(a > 0\), both sides are positive, and we can take logarithms (Theorem 4):

\[ \begin{aligned} a^{bc} &= a^{(b^c)} \\ {\operatorname{log}\mathopen{}\left\{a^{bc}\right\}\mathclose{}} &= {\operatorname{log}\mathopen{}\left\{a^{(b^c)}\right\}\mathclose{}} && \text{(take logarithms of both sides)} \\ bc \cdot {\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} &= b^c\cdot {\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} && \text{(logarithm of a power)} \end{aligned} \tag{2}\]

The last line of Equation 2 holds exactly when

1. \(a = 1\) (so that \({\operatorname{log}\mathopen{}\left\{a\right\}\mathclose{}} = 0\)), or
2. \(bc = b^c\) (see Theorem 10).

Case \(a = 0\). Here we cannot take logarithms, so we work from the values of powers of \(0\): \(0^s = 0\) for \(s > 0\) (Definition 26), \(0^0 = 1\), and \(0^s\) is undefined for \(s < 0\).

• If \(b < 0\), then \(0^b\) is undefined, so \((a^b)^c\) is undefined.

• If \(b > 0\), then \((0^b)^c = 0^c\) and \(b^c > 0\), so \(0^{(b^c)} = 0\); the two sides agree exactly when \(c > 0\).

• If \(b = 0\), then

  \[ \begin{aligned} (0^0)^c &= 1^c \\ &= 1; \end{aligned} \]

  for \(c > 0\),

  \[ \begin{aligned} 0^{(0^c)} &= 0^0 \\ &= 1, \end{aligned} \]

  so the two sides agree, and for \(c \le 0\) they do not.

So for \(a = 0\), \((a^b)^c = a^{(b^c)}\) exactly when \(b \ge 0\) and \(c > 0\).

In particular, when \(a = 0\), \(b \ge 0\), and \(c = 0\), the two sides differ:

\[ \begin{aligned} (a^b)^c &= (0^b)^0 \\ &= 1 \end{aligned} \]

\[ \begin{aligned} a^{(b^c)} &= 0^{(b^0)} \\ &= 0^1 \\ &= 0 \end{aligned} \]

LemTheorem 11 (When repeated exponentiation is not reordered)

Theorem 11 (When repeated exponentiation is not reordered) Let \(a \ge 0\) and \(b, c \in \mathbb{R}\), where \(b^c\) is defined. Then \((a^b)^c = a^{(b^c)}\), with both sides defined, if and only if

• \(a > 0\) and either \(a = 1\) or \(bc = b^c\); or
• \(a = 0\), \(b \ge 0\), and \(c > 0\).

ProofProof

Proof. This is the solution to Exercise 3, worked in Solution 3.

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