Lecture 5  Logarithms, Slope of a Function, Derivatives
5.1 Logarithms
Note the graph of
e
x
This graph passes the horizontal line test, so
f
(
x
) =
e
x
is onetoone and therefore has an
inverse function. This is also true of
f
(
x
) =
a
x
for any
a >
0
,a
6
= 1.
More generally, for any
a >
1 the graph of
a
x
and its inverse look like this. If
f
(
x
) =
a
x
,
then we deﬁne the inverse function
f

1
to be the
logarithm with base
a
, and write
f

1
(
x
) = log
a
(
x
)
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View Full DocumentNote that, since the image of
a
x
is only the positive numbers, the domain of log
a
(
x
) is all
positive real numbers. The key property is:
log
a
x
=
b
⇐⇒
a
b
=
x
Examples:
log
10
10 = 1
10
?
= 10
log
5
25 = 2
5
?
= 25
log
4
1
2
=

1
2
4
?
=
1
2
log
5
1
125
=

3
5
?
=
1
125
↑
↑
log equation
corresponding
exponential equation
Log Rules
1. Most important: by the properties of inverse functions we have
log
b
(
b
x
) =
x
and
b
log
b
x
=
x
The most important case of logs is when
b
=
e
. Log base
e
has a special name, in fact
we deﬁne log
e
x
= ln(
x
). So the above becomes
ln(
e
x
) =
x
and
e
ln(
x
)
=
x
LEARN THIS!!
The function ln(
x
) is known as the
natural logarithm function
, and ln(
x
) should
be read as “the natural logarithm of
x
”. In class, you may also hear me read this as
“lawn
x
”, but this isn’t as standard.
Other rules: (I will state for ln, but they work for every log). Suppose that
x,y >
0
2. ln(
xy
) = ln(
x
) + ln(
y
)
3. ln
±
x
y
²
= ln
x

ln
y
4. ln(
x
y
) =
y
ln(
x
)
Calculations:
e
3ln(
x
)
=
e
ln(
x
3
)
=
x
3
ln
³
1
e
´
= ln(
e

1
) =

1
Rewrite the following:
ln
±
xy
z
²
= ln(
xy
)

ln(
z
) = ln(
x
) + ln(
y
)

ln(
z
)
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 Fall '09
 PIETERHOFSTRA
 Derivative, Slope, 5%, Logarithm, $1000, 0 g, 13.9 years

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