X.
EXPONENTIALS
AND
LOGARITHMS
1.
The
Exponential and
Logarithm Functions.
We have so far worked with the algebraic functions

those involving polynomials and
root extractions

and with the trigonometric functions. We now have to add to our list
the
exponential and logarithm functions, since these are used in your science and engineering
courses from the beginning. Your book will handle the calculus
of
these functions; here we
want to review briefly their algebraic properties, and look at applications; one or two of
them might be new to you.
Where does one encounter exponentials and logarithms? In general, exponentials are
used to express all sorts of simple growth and decay processes.
1.
The growth of a
bacteria colony which doubles in
size every day:
y = yo2
t ,
where t
=
time in days, y
=
population size, yo
=
the initial size, i.e., size at
t
= 0.
2. Dollars in a bank account, at
5%.interest compounded annually:
A
=
Ao(l.05)
n ,
where
n
=
number of years,
A
=
amount, Ao
=
initial amount (the "principal").
3.
Amount of radioactive substance, with a
1
year halflife:
x
=
zo(1/2)n
=
zo2

n,
where
n
=
number of years,
x
=
amount,
0o=
initial amount.
There are many other examples: the decay of electric charge on a capacitor, the way
a hot and cold body come to the same temperature when they are brought together,
the
falling of a body through a resisting medium (a steel ball dropped into oil, for instance)

all involve exponentials when you express them in mathematical terms.
We use
logarithms when
the base of the exponential is unimportant and we want to focus
our attention on the exponents instead. Suppose the base is
10;
writing simply "log" for
"log
0
o"
in what follows, we have
y
= 10'
€
logy =
z.
1.
Star
magnitude. The observed brightness
B
of a star is described
by
comparing it
to a standard brightness
Bo,
using the equation
B
=
BolO

5
.
10
The important thing is the number
m
called the
magnitude;
it is defined
by
the above
equation, or equivalently, taking logs,
by
5
B
m
= 
log
.
2