The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

(26)
That is, the log of 
e
to itself as a base is 1 (as is the log of 10 to the base 10).
6.
ln 1 
0
(27)
That is, the natural log of the number 1 is zero (as is the common log of number 1).
7. If 
Y
ln 
X
,
dY
d X
=
1
X
(28)
That is, the rate of change (i.e., the derivative) of 
Y
with respect to 
X
is 1 over 
X
. The exponential
and (natural) logarithmic functions are depicted in Figure 6A.1.
Although the number whose log is taken is always positive, the logarithm of that number can be
positive as well as negative. It can be easily verified that if
0
<
Y
<
1
then
ln
Y
<
0
Y
=
1
then
ln
Y
=
0
Y
>
1
then
ln
Y
>
0
Also note that although the logarithmic curve shown in Figure 6A.1(
b
) is positively sloping,
implying that the larger the number is, the larger its logarithmic value will be, the curve is increasing
at a decreasing rate (mathematically, the second derivative of the function is negative). Thus, ln(10)
=
2.3026 (approx.) and ln(20) 
=
2.9957 (approx.). That is, if a number is doubled, its logarithm does
not double.
This is why the logarithm transformation is called a nonlinear transformation. This can also be
seen from Equation (28), which notes that if 
Y 
= 
ln 
X
, 
dY
dX
=
1
X
. This means that the slope of the
logarithmic function depends on the value of 
X
; that is, it is not constant (recall the definition of
linearity in the variable).
Logarithms and percentages:
Since 
d
(ln
X
)
d X
=
1
X
, or 
d
(ln
X
)
=
d X
X
, for very small changes the
change in ln 
X
is equal to the relative or proportional change in 
X
. In practice, if the change in 
X
is
reasonably small, the preceding relationship can be written as the change in ln
X
≈
to the relative
change in 
X
, where 
≈
means approximately.
FIGURE 6A.1
Exponential and
logarithmic functions:
(
a
) Exponential
function; 
(
b
) logarithmic
function.
(
a
)
1
1
0
0
45
°
Y
(
b
)
X 
=
 
ln
 Y
X 
=
 
ln
 Y
Y
X
Y
= 
e
X
45
°
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