\documentclass[11pt]{article}
\usepackage{precalc}
\begin{document}
\assigntitle{29}{Logarithm review}
% Logarithms!
%
% Logarithms are the reverse of exponentiation.
%
% Definition. Examples.
% Lots of exercises.
%
% Translate.
% Find the log.
%
% Log rules (derive).
% * addition/subtraction.
% * "slap down" rule.
% * change of base.
%
% Use logs to solve equations.
%
% What's the big deal?
% 1. Comes up a lot in calculus! In particular the natural
% logarithm.
% 2. Turns multiplication into addition. Used to be used for
% computation/slide rules etc, although not so much anymore.
\topic{review?}
This week we're going to spend some time reviewing logarithms. I say
``review'' since you've probably seen them before in theory, but if my
experience is any guide, it's quite likely that you've forgotten most
of what you used to know about them!
\topic{why students dislike logarithms}
In my experience, most students seem to really despise logarithms.
I'm not entirely sure why, but I have a few guesses:
\begin{enumerate}
\item Learning about logarithms often seems to consist of just
learning a bunch of (seemingly) arbitrary rules, and using them to
solve (tedious, uninteresting) problems.
\item They never seem to come up in any other topic---they're just
like some isolated topic that you learn for no good reason, and
never hear about again!
\item The arbitrary rules alluded to in item (1) are confusing and
difficult to remember.
\end{enumerate}
\begin{problem}
Sound familiar? With which of these items do you agree? Are there
any other reasons? (If you think logarithms are happy fun times, it's
OK to say that too.)
\end{problem}
\topic{the real scoop}
I also have some responses to these assertions:
\begin{enumerate}
\item There's actually only \emph{one} rule that you need to
know---all the other rules follow from it, if you understand that
one rule really well. I can't really argue about the tediousness of
the sorts of problems that use logarithms, unfortunately---but see
the next response.
\item It turns out that logarithms come up \emph{all the time}, in
some very fundamental ways, in the study of calculus. Now, there
\emph{used} to be a very good reason to learn about logarithms long
before you got to calculus---which I will explain later. The
problem is that \emph{this reason no longer exists}, so no wonder
students feel like they are an isolated topic with no relation to
anything else---because until you get to calculus, it's true!
So I hope you can trust me when I say: understanding logarithms
\emph{will} be useful, eventually, and if you feel like they seem
kind of pointless now, you are not wrong.
Logarithms (specifically, $\log_2$) also come up a lot in the branch
of computer science that studies algorithmic complexity.
\item Arbitrary rules are only difficult to remember if you don't use
them a lot. See point (2).
\end{enumerate}
With that out of the way---onwards!
\section{Logarithms and exponents}
\topic{the only thing you need to know}
Here is the most important---in some sense, the \emph{only}---thing
you need to know: logarithms are the opposite of exponents! More
specifically:
\begin{defn}{logarithms}
If \[ b^e = a \] then \[ \log_b a = e. \] (Note: to typeset
logarithms in \LaTeX, use \verb|\log|: for example,
\verb|\log_6 (q+1)| renders as $\log_6 (q+1)$.
\end{defn}
Put another way: \[ b^e = ? \] asks, ``if you multiply $b$ by itself $e$
times, what do you get?'' and \[ \log_b a = ? \] asks, ``how many times do
you have to multiply $b$ by itself to get $a$?''
\topic{translation exercise}
\begin{problem}
Translate each exponential equation to an equivalent one using logarithms,
and vice versa.
\begin{subproblems}
\item $2^8 = 256$
\item $3 = \log_b 125$
\item $z = x^{1024}$
\item $\log_8 q = f$
\end{subproblems}
\end{problem}
\begin{problem}
Evaluate:
\begin{subproblems}
\item $\log_2 16$
\item $\log_5 5$
\item $\log_4 64$
\item $\log_9 3$
\item $\log_7 1$
\end{subproblems}
\end{problem}
\begin{problem} \label{prob:log-special}
\topic{some special logarithms}
Suppose $b > 1$.
\begin{subproblems}
\item What is $\log_b b$?
\item What is $\log_b 1$?
\item What is $\log_b b^e$?
\item What is $b^{\log_b e}$?
\end{subproblems}
\end{problem}
\section{Logarithm rules}
\label{sec:rules}
\topic{logarithm rules}
There are three main rules specifying how logarithms can be
manipulated. However, each of them is a direct consequence of the
definition of logarithms from the previous section (as the inverse of
exponentiation). Let's see if you can figure them out.
\begin{problem} \label{prob:log-addition}
\topic{the multiplication/addition rule}
Consider $\log_b (x y)$.
\begin{subproblems}
\item I claim that $x = b^{\log_b x}$. Why is this?
\item Of course, $y = b^{\log_b y}$ as well. Substitute these two
expressions for $x$ and $y$ in the expression $\log_b (xy)$.
What do you get?
\item Can you simplify the resulting expression, using the laws of exponents?
\item Can you simplify the result again, using what you know about
logarithms? (Hint: see \pref{prob:log-special}\dots)
\item What logarithm law have you discovered?
\end{subproblems}
\end{problem}
In English, this law says that \emph{the logarithm of a product is the
sum of the logarithms}. In other words, logarithms turn
multiplication into addition!
\topic{the division/subtraction rule}
It is likewise true (although I won't make you show this one; it is
quite similar to \pref{prob:log-addition}) that logarithms turn
division into subtraction: \[ \log_b (x/y) = \log_b x - \log_b y. \]
\emph{This} is why logarithms were once useful outside of calculus:
\emph{adding} is a lot easier than \emph{multiplying}, so logarithms
could be used to help perform multiplication much more quickly.
Here's how it worked: say you wanted to multiply $x$ and $y$, which
are too big to easily multiply by hand.\footnote{Note, when I say
\emph{big}, I really just mean \emph{having a lot of decimal
places}: it is just as tedious to multiply $1.23456789$ by
$89.362349763$ as it is to multiply $123456789$ by $89362349763$.}
So you get out your handy Table O' Logarithms\footnote{By which I mean
Enormous Book O' Logarithms.} and look up the logarithms of $x$ and
$y$. Then you add those (which is pretty easy) and get $\log_e x +
\log_e y$ (note that most Tables O' Logarithms were to the base
$e$). But $\log_e x + \log_e y = \log_e (xy)$, so now you take this
number and do a reverse lookup (in the second half of your Enormous
Book---kind of like a bilingual dictionary) to see what it is the
logarithm \emph{of}, and of course you get $xy$.
\begin{problem}
Why doesn't anyone have an Enormous Book O' Logarithms anymore?
\end{problem}
\begin{problem}
This same idea was the basis for \emph{slide rules}. Look up slide
rules on the Internet (Wikipedia is a good starting place, but also
try following some of the ``related links'' at the bottom of the
page) and explain what they were, how you used them, and why no one
really uses them anymore.
\end{problem}
\begin{problem}
If logarithms turn multiplication into addition, then they turn
exponentiation into\dots what? (\emph{Hint:} think about $\log_b
(x^a)$. What does $x^a$ mean? Can you apply the
multiplication-to-addition rule?)
\end{problem}
\begin{problem}
Simplify.
\begin{subproblems}
\item $\log_2 (4^7)$
\item $\log_3 (x^8 9^2)$
\item If $\log_b 3 = 1.4$, what is $\log_b 27$?
\item If $\log_b 5 = x$ and $\log_b 3 = y$, what is $\log_b 225$?
\end{subproblems}
\end{problem}
\topic{change-of-base formula}
And now for the final rule: the \term{change-of-base} formula. \[
\log_a x = \frac{\log_b x}{\log_b a}. \] This says that the logarithm
to base $a$ of $x$ is the same as the logarithm to base $b$ of $x$,
divided by the logarithm to base $b$ of $a$. This is a very useful
formula to know for evaluating logarithms on your graphing calculator,
since it can only do logarithms base $10$ and base $e$; you can use
the change-of-base formula to evaluate a logarithm to any base $a$ as
long as you can evaluate logarithms to some particular base $b$ (with
your calculator, $b = 10$ or $e$).
\begin{problem}
You graphing calculator has two buttons for performing logarithms.
The button labelled ``log'' does $\log_{10}$. The button labelled
``ln'' (which is an abbreviation for ``natural logarithm'' (probably
in French or something)) does $\log_e$. What is $e$? Well, it's
approximately $2.71828\dots$ but you'll have to wait until calculus
to find out why it's so special!
Use your graphing calculator to evaluate each of the
following. Round your answers to three decimal places.\footnote{Or
whatever.}
\begin{subproblems}
\item $\log_2 50$
\item $\log_{10} 200$
\item $\log_9 27$
\end{subproblems}
\end{problem}
\begin{problem}
Use your graphing calculator to make a graph of $y = \ln x$.
Describe the graph. Give as much detail as possible.
\end{problem}
\begin{problem}
Solve for $x$.
\begin{subproblems}
\item $2^{x+5} = 4^x$
\item $5^{x - 3} = 17$
\item $\log_7 (3x) = 5$
\item $5^x = 3^{2x+1}$
\end{subproblems}
\end{problem}
\end{document}