[IX : 202]
THE LOGARITHMIC AND EXPONENTIAL FUNCTIONS
450
202.
Scales of infinity.
The logarithmic scale.
Let us consider
once more the series of functions
x,
√
x,
3
√
x, . . . ,
n
√
x, . . . ,
which possesses the property that, if
f
(
x
) and
φ
(
x
) are any two of the functions
contained in it, then
f
(
x
) and
φ
(
x
) both tend to
∞
as
x
→ ∞
, while
f
(
x
)
/φ
(
x
)
tends to 0 or to
∞
according as
f
(
x
) occurs to the right or the left of
φ
(
x
) in
the series. We can now continue this series by the insertion of new terms to the
right of all those already written down. We can begin with log
x
, which tends to
infinity more slowly than any of the old terms. Then
√
log
x
tends to
∞
more
slowly than log
x
,
3
√
log
x
than
√
log
x
, and so on. Thus we obtain a series
x,
√
x,
3
√
x, . . . ,
n
√
x, . . .
log
x,
p
log
x,
3
p
log
x, . . .
n
p
log
x, . . .
formed of two simply infinite series arranged one after the other. But this is not
all. Consider the function log log
x
, the logarithm of log
x
. Since (log
x
)
/x
α
→
0,
for all positive values of
α
, it follows on putting
x
= log
y
that
(log log
y
)
/
(log
y
)
α
= (log
x
)
/x
α
→
0
.
Thus log log
y
tends to
∞
with
y
, but more slowly than any power of log
y
.
Hence we may continue our series in the form
x,
√
x,
3
√
x, . . .
log
x,
p
log
x,
3
p
log
x, . . .
log log
x,
p
log log
x, . . .
n
p
log log
x, . . .
;
and it will by now be obvious that by introducing the functions log log log
x
,
log log log log
x
, . . . we can prolong the series to any extent we like. By putting
x
= 1
/y
we obtain a similar scale of infinity for functions of
y
which tend to
∞
as
y
tends to 0 by positive values.