In doing the exercises related to this section of the module (and in general), you will
need to use the “
y
x
” or “
x
y
” button on your scientific calculator. Typically, the sequence
of commands for, say, “2
3
= ?” is [2
➔
“
x
y
”
➔
3
➔
=], but if you haven’t used this button
before, you should experiment with some simple examples, before doing the exercises.
Logarithms and Exponential Functions:
If
b
x
= a
, then
x
is the logarithm of
a
to base
b
, that
is, the power to which
b
must be raised to give
a
: log
b
a
=
x.
The two most common bases
for logarithms are 10 and
e
. Logarithms to these two bases are written “log” (or log
10
)
and
ln
(for “natural” logarithm), respectively. Recall that
e
is a transcendental number
(with a value of approximately 2.71828) defined as the limit of the expression (1 + 1/
n
)
n
as
n
approaches infinity, a formula which represents
instantaneous compounding
of a
growth rate, as
n
, the number of subperiods within a period, becomes infinite and the
growth rate per subperiod becomes infinitesimal.
It is possible to convert logarithms from one base to another. If
b
x
=
a
y
, then
x
log
b
b
= y
log
b
a
, and since the logarithm of any number to itself as base (in this case, log
b
b
) =
1, we have
x = y
log
b
a
. Your scientific calculator has log
10
, 10
x
,
ln
and
e
x
functions. You
may want to confirm by punching 10 into your calculator and hitting the
ln
button that
log
e
10 =
ln
10 = approximately 2.302585. Hence if log
10
a
=
x,
then log
e
a
= 2.302585
x
.
Using our knowledge of exponents, we know immediately that log
10
10 = 1, log
10
100
= 2, log
10
1 = 0, and log
10
0.1 = log
10
1/10 = –1. We also know that numbers between 10
and 100 have logarithms between 1 and 2, and we trust our calculators to remember
precisely what they are.
We shall mention only 2 applications of logarithms and exponents here. First, using
a result from Module 10, we have that for the function
ln x
, the derivative
d
(
ln x
)/
dx
=
1/
x, and so d
(
ln x
)
= dx/x
. Since the same relation holds true for a variable