Graphics Reference
In-Depth Information
17 Find a positivenumbr
M
such that
a
a
x
y
M
,
provided 0
y
x
c
.
Real numbers as indices
18 How does the condition in qn 17 guarantee that
A
may be
extended in a unique way to a continuous function on [0,
)?
19 How can you besurethat
A
may also be extended in a unique way
to a continuous function on (
, 0]?
This gives us a continuous function
A
:
R
R
, and at last wehave
a well-defined meaning for irrational indices.
20 For
a
1 and
x
,
y
R
, provethetwo laws of indics
(i)
a
a
a
and (ii)
a
(
a
)
.
For (ii), first establish the result for
y
n
N
, then for
y
1/
n
, and
then for
y
m
/
n
Q, before attempting
y
R.
21 Show that
A
is strictly increasing on
R
.
Since
A
: R
R
is strictly increasing it is a bijection and has a
unique inverse:
R
R
called the logarithm to the base
a
.
A
(
x
)
log
x
.
22 Provethat, when
X
,
Y
0, log
X
log
Y
log
XY
.
Deduce that log
1/
X
log
X
.
23
0, check that
D
is continuous at
x
if and only if
A
is
continuous at
x
. See qn 12.
(ii) Usetheargument of qn 21 to show that
D
is strictly
increasing on R
.
(iii) If (
x
(i) For
x
) is a null sequence of positive terms, and
q
is a rational
number lying between
x
and 2
x
, usethefact that
(
D
(
q
))
L
(
a
) and (
D
(
q
))
L
(
a
) to provethat
D
(
x
)
L
(
a
).
(iv) By an argument like that of qn 15 show that
lim
(
D
(
x
))
L
(
a
) and deduce that lim
D
(
x
)
L
(
a
).
(v) If
D
(0)
L
(
a
), must
D
becontinuous on R?
24 Provethat
A
(
x
)
A
(
x
)ยท
L
(
a
) for
x
R
.
