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series. This enabled him to construct the series for exp(
x
)
1. Newton
did not use the terminology of logarithms or exponentials. In 1668
Mercator published the series for log(1
Logarithmotechnia
and referred to areas under the rectangular hyperbola as
natural
logarithms. Leibniz described the integral
x
)inhis
dx
/
x
as a logarithm in 1676.
Negative and fractional exponents were first proposed by Wallis in
1656 and first used by Newton in his discussion of theBinomial
Theorem in 1669. The first overt claim of the connection between
exponents and logarithms was made by Wallis in 1685 in relation to
the terms of a geometric progression.
Thefirst claim that thecommon logarithm of
x
was thepowr to
which 10 had to beraised to obtain
x
was made in tables published in
England in 1742. In Euler's
Introductio in Analysin Infinitorum
published in 1748, he explicitly defined log
x
y
where
a
x
. Heused
infinitesimals and infinite numbers to great effect. For an infinitesimal
,
helt (
a
k
(which depends on
a
) and called the value of
a
which makes
k
1, e (the first letter of his name). By combining
infinitesimal and infinite numbers
N
x
, heobtained
a
as the
exponential series exp(
kx
). The conventional exponential series follows.
Theoriginal equation giving
k
, was rearranged to give (1
x
/
N
)
e
.
Further ingenious manipulation led to the result that
1)/
log
x
(
N
/
k
)(
x
1),
and he then used the binomial series to obtain the series for log(1
x
).
In 1821, Cauchy, presuming that
a
was well defined, proved that
every continuous function
f
satisfying
f
(
x
y
)
f
(
x
)ยท
f
(
y
)
was of theform
f
(
x
)
a
for some
a
. Heused theCauchy product to
show that the exponential power series belonged to this family of
functions, and defining e
1/
n
! had thereby proved that the
exponential power series was equal to e
. Cauchy also used the
Binomial Theorem to prove that, as
a
0, (1
ax
)
e
. In 1881, A.
Harnack defined
a
for rational
x
and extended his definition to
irrational
x
by aMrming thecontinuity of thenew function.
In order to extend exponentials to complex numbers, Euler (1748)
had defined exp(
z
) by means of the exponential series 1
z
z
/2!
...
and in the late nineteenth century it was recognised that, if this
definition was used for a real variable, a formal definition of
a
as
exp(
x
log
a
) avoids someof thediMcultis with which Cauchy had
struggled.
Newton's discovery of the sine series depended on his previous
discovery of the Binomial Theorem for rational index. He derived the
