Exponential and logarithm functions
The graph of
=
y
2
x
Let f be defined by
=
( )
f
x
2
x
.
We compute some values of f in order to sketch its graph.
=
( )
f 1
2
( )
1
=
2,
=
( )
f 2
2
2
=
4,
=
( )
f 3
2
3
=
8,
=
( )
f 0
2
0
=
1,
=
(
)
f

1
2
(
)

1
=
1
2
=
0.5,
=
(
)
f

2
2
(
)

2
=
1
2
2
=
1
4
=
0.25,
=
(
)
f

3
2
(
)

3
=
1
2
3
=
1
8
=
0.125,
=
f
1
2
2
1
2
=
2 ~
1.41421,
=
f
3
2
2
.
2
1
2
=
2
2 ~
2.82843,
=
f
5
2
2
5
2
=
2
2
.
2
1
2
=
4
2 ~
5.65685,
=
f

1
2
2

1
2
=
1
2
1
2
=
1
2
=
2
2
~
0.707107,
=
f

3
2
2

3
2
=
1
2
3
2
=
1
2
2
=
2
4
~
0.353553,
=
f

5
2
2

5
2
=
1
2
5
2
=
1
4
2
=
2
8
~
0.176777
These values are collected together in the following table.
x


3

5
2

2

3
2

1

1
2
0
1
2
1
3
2
2
5
2
3
( )
f
x

1
8
2
8
1
4
2
4
1
2
2
2
1
2
2
2
2
4
4
2
8

0 .125000
0 .176777
0 .250000
0 .353553
0 .500000
0 .707107
1
1.41421
2
2.82843
4
5.65685
8
Joining the points given by the table to form a continuous (unbroken) curve amounts to suggesting that the domain of the function f is the
set of all real numbers.
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View Full DocumentIn order to justify this first consider what
2
x
means if
x
is a
rational number
.
For a positive rational number
=
x
p
q
, where
p
and
q
are
positive integers,
=
( )
f
x
2
p
q
=
q
2
p
, that is, the
q
th
root of 2
p
.
For example,
=
(
)
f 1.6
=
f
8
5
2
8
5
=
5
2
8
=
5
256 ~
3.03143.
For a negative rational number
=
x

p
q
, where
p
is a negative integer and
q
is a positive integer,
=
( )
f
x
2

p
q
=
1
2
p
q
= 1/
q
2
p
.
For example,
=
(
)
f

1.6
=
f

8
5
2

8
5
= 1/
5
2
8
~
0.329877.
For a general real number
x
we can think of
x
being being approximated sufficiently accurately by a rational number in order to calculate
=
( )
f
x
2
x
, to a desired degree of accuracy. For example,
we can imagine
2
π
being computed to 6 figure accuracy by using a 7 figure
rational approximation for
π
, namely
=
3.141593
3141593
1000000
.
Then 2
π
~
1000000
2
3141593
~
8.82498. Actually, this is a very inefficient way
to perform the computation if all the digits of
2
3141593
are obtained since there are 945714 of them.
However, this example is given in an
attempt to explain why the
domain
of the function f where
=
( )
f
x
2
x
is indeed the set of
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 Spring '08
 Uri
 Logarithm, loga, loga b

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