Skip to main content
Contents Index Dark Mode Prev Up Next \(\usepackage{siunitx}
\newcommand{\lrp}[1]{\left(#1\right)}
\newcommand{\lrb}[1]{\left[#1\right]}
\newcommand{\lrbrace}[1]{\left\lbrace#1\right\rbrace}
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\dint}{\displaystyle\int}
\newcommand{\defint}[2]{\dint^{#2}_{#1}}
\newcommand{\dlim}[2]{\displaystyle\lim_{#1\rightarrow #2}\,}
\newcommand{\dydx}{\dfrac{dy}{dx}}
\newcommand{\ddx}{\tfrac{d}{dx}}
\newcommand{\dddx}{\dfrac{d}{dx}}
\newcommand{\ifsol}[1]{\ifprintanswers{#1}\fi}
\newcommand{\Nat}{\mathbb{N}}
\newcommand{\Whole}{\mathbb{W}}
\newcommand{\Int}{\mathbb{Z}}
\newcommand{\Rat}{\mathbb{Q}}
\newcommand{\Real}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\DeclareMathOperator\arcsinh{arcsinh}
\DeclareMathOperator\arccosh{arccosh}
\DeclareMathOperator\arctanh{arctanh}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\newcommand{\sfrac}[2]{{#1}/{#2}}
\)
Section 1.16 Sweet 16
Subsection 1.16.1 Recommended Tutorials
Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
Subsection 1.16.2 Introduction
We all know that \(16\) exhibits the following interesting property:
\begin{equation*}
16 = 2^4 = 4^2.
\end{equation*}
In this activity, you will explore whether there exist any other positive integers \(a\) and \(b\) (\(a \not = b\) ) such that
\begin{align}
a^b \amp = b^a\text{.}\tag{1.5}
\end{align}
To determine if there are integer solutions to equation
(1.5) , you will need to rely on unusual tactics, including the use of Rolle’s Theorem:
Theorem 1.8 . Rolle’s Theorem.
Let \(f\) be a function that satisfies the following three hypotheses:
\(f\) is continuous on the closed interval \([a,b]\text{.}\)
\(f\) is differentiable on the open interval \((a,b)\text{.}\)
\(f(a) = f(b)\text{.}\)
Then there is a number \(c\) in \((a,b)\) such that \(f'(c) = 0\text{.}\)
Aside If we remove the restriction that
\(a\) and
\(b\) have to be integers, then this problem becomes a lot more interesting.
It can be shown that for any value \(N > { e}^{ e}\text{,}\) there exist real numbers \(a\) and \(b\) that satisfy \(N = a^b = b^a\) with \(a \neq b\text{.}\) To prove this, one needs to be very careful in evaluating the limit
\begin{equation*}
\lim_{\substack{a \rightarrow 1\\ b \rightarrow \infty} } a^b\text{.}
\end{equation*}
It is not obvious that this limit goes to infinity. However, if it does, all \(N > { e}^{ e}\) will have the desired property. It may help to recall that
\begin{equation*}
\lim_{n \rightarrow \infty} \left(1 + \frac{1}{n} \right)^n = { e}\text{.}
\end{equation*}
So, just because the base tends to \(1\) and the exponent tends to infinity, the value of \(a^b\) is not necessarily infinity.
Exercises 1.16.3 Exercises
1.
If you rearrange equation
(1.5) to separate variables, you can write the equation as
\begin{equation*}
\frac{\ln(a)}{a} = \frac{\ln(b)}{b}\text{.}
\end{equation*}
You can obtain this equivalent equation by taking the natural logarithm of both sides of equation
(1.5) . Then, divide that result by both
\(a\) and
\(b\text{.}\) Notice how the equation in this form is set up so that you can apply Rolle’s Theorem.
(a) Assign the function
\(f(x) = \frac{\ln(x)}{x}\) in Maple.
(b) Evaluate the derivative of
\(f(x)\text{.}\) Notice that
\(f\) is differentiable (and therefore continuous) for all
\(x > 0\text{.}\)
(c) Solve for any critical value(s) of
\(f(x)\) where
\(f'(x) = 0\text{.}\) How many critical values are there?
2.
Now you are ready to apply Rolle’s Theorem (and a bit of reasoning) to determine if there are any other positive integers
\(a\) and
\(b\) that satisfy equation
(1.5) . There are no calculations to perform in this exercise. Instead, you will need to adequately explain your answers in paragraph form.
Let’s begin by assuming that
\(a \lt b\text{.}\)
(a) If
\(a\) and
\(b\) satisfy equation
(1.5) , then
\(f(a) = f(b)\text{.}\) From what Rolle’s Theorem states, what does this tell you about the location of a critical value of
\(f(x)\) where
\(f'(x) = 0\text{?}\)
(b) Since you have already found the critical value(s) of
\(f(x)\text{,}\) can there be any other positive integers
\(a\) and
\(b\) such that
\(f(a) = f(b)\text{?}\)
3. What have you concluded about the problem? Are there any other positive integers
\(a\) and
\(b\) (
\(a \not = b\) ) that satisfy equation
(1.5) ?
