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in fact theorems. It will be necessary to establish that these two laws of
indices still hold as we extend the family of numbers which may be used
as indices.
2 For
m
,
n
N, provethat when
x
1,
m
n
x
x
, from order
properties in qn 2.11.
Deduce that, when 0
x
1,
m
n
x
x
.
3 From qn 6.28, show that, for
n
N, thefunction
f
: R
R
given by
f
(
x
)
x
is strictly increasing, continuous and
unbounded above.
Positive rationals as indices
4 With
f
as in qn 3, show that
f
exists by the
Intermediate Value Theorem, and is strictly increasing, continuous
and unbounded above.
:
R
R
f
(
x
) (as in qn 4) is denoted by
x
.
For
x
0 and
n
,
m
Z
,
x
(
x
)
.
5 (i) For
x
,
y
0, provethat (
xy
)
x
y
.
(ii) By induction on
n
provethat
x
(
x
)
.
(iii) Show that
x
(
x
)
. Use(ii) and thefact that the
function
x
x
is a bijection of R
.
6 For
x
, provethetwo laws of indics
(i)
x
x
x
and (ii)
x
(
x
)
. Build on qns 1 and 5.
0 and
r
,
s
Q
7 Provethat, when
x
.
Deduce that, when 0
x
1, and
r
,
s
Q
,
r
s
x
x
.
1, and
r
,
s
Q
,
r
s
x
x
Rational numbers as indices
For
x
0 and
r
Q
,
x
1 and
x
1/
x
.
8 For
x
0 and
r
,
s
Q, provethetwo laws of indics
(i)
x
x
x
and (ii)
x
(
x
)
.
9 Provethat, when
x
1, and
r
,
s
Q,
r
s
x
x
.
Deduce that, when 0
x
1, and
r
,
s
Q
,
r
s
x
x
.
For
a
1, define
A
:
Q
R
by
A
(
x
)
a
.
