Graphics Reference
In-Depth Information
11
Indices and circle functions
For an elegant treatment of exponential and logarithmic functions
starting from the counter-intuitive definition log
x
dt
/
t
see Hardy,
or for a presentation of Hardy's treatment using a problem sequence
see Quadling. For an alternative treatment starting from thedefinition
x
2!
x
3!
...
exp(
x
)
1
x
see Burkill.
Exponential and logarithmic functions
Although we have used logarithms and exponentials in this topic,
they have not played a formal roˆ le in defining any of the concepts of
analysis.
We now have the results we need to define these functions and to
establish their properties.
Positive integers as indices
For all
x
R,
x
x
and
x
x
x
;
n
N.
For
x
0,
x
1 and
x
1/
x
.
1 For
m
,
n
N, proveby induction on
n
that
,
(ii)
x
(
x
)
, and
(iii) (
xy
)
x
y
.
(i)
x
x
x
Properties (i) and (ii) are called the
laws of indices
though they are
