74
II - Convergence: Discrete variables
(or rv Mh where M is a constant if one works, like Briggs, with logarithms to
base 10), in other words, that the function log has a derivative l3 equal to 1
at x = 1.
In these circumstances, 10g(1 + xn)/xn tends to 1 by definition of the
derivative, thus log(x)/nxn also, and, since logx does not depend on n, one
sees finally that log x = lim nxn. Taking account of the definition (6) of x n ,
one thus obtains the fundamental formula
(5.7)
log x = limn(xl/n -1).
This seems to be due to the astronomer Halley (1695), although the essential
ideas are already in Napier and Briggs.
Exercise. Show that (7) is equivalent to
lin 1 log x
( 1 )
x
= + - - +0 -
n
n
as n ~ +00.
To avoid all confusion, we stress the fact that, at this stage of the exposition, the formula (7) is not, in itself, either a definition or a construction
of the function log. We have only shown that, if there exists a function log
satisfying log(xy) = log x + logy and such that log(l + x) rv X when x tends
to 0, then it is given by the relation (7). But we have as yet proved neither
the existence of the function log, nor the convergence of the sequence (7).
This will be the object of Theorem 3 of nO 10.
In fact, as we shall see later, 10g(1 + u) lies, for 0 < u < 1, between
u - u 2 /2 and u. Since
log x = n.log(l + x n ),
we have
(5.8)
o < nXn - log x < nx;'/2 < x 2 /2n,
the last inequality following from (6). In other words, the error committed in
replacing log x by nXn is < x 2 /2n provided that Xn < 1. It only remains, a
modest enterprise, to perform the numerical calculations.
13 This is the crucial point in obtaining formula (7). Napier, who worked at his
tables from about 1590 to his death in 1617, and Briggs, who transformed them
into logarithms to base 10 from about 1615 on, did not argue in terms of "derivatives" for the excellent reason that these would not appear, and then in a rather
hazy way, until about twenty years later, with Fermat and Descartes, Ii propos
the calculus of tangents to a curve. But Napier imagined a point moving along
a segment of line with a speed inversely proportional to its distance x from the
origin of the segment, and the concept of "instantaneous speed" was just that of
derivative with respect to time. The reader should not believe that the concepts
and ideas which, nowadays, appear simple enough to be taught every year to
hundreds of thousands of young people of the Earth, were born fully armed from
certain brains of genius, like Athena from that of Jupiter ...
II - Convergence: Discrete variables
(or rv Mh where M is a constant if one works, like Briggs, with logarithms to
base 10), in other words, that the function log has a derivative l3 equal to 1
at x = 1.
In these circumstances, 10g(1 + xn)/xn tends to 1 by definition of the
derivative, thus log(x)/nxn also, and, since logx does not depend on n, one
sees finally that log x = lim nxn. Taking account of the definition (6) of x n ,
one thus obtains the fundamental formula
(5.7)
log x = limn(xl/n -1).
This seems to be due to the astronomer Halley (1695), although the essential
ideas are already in Napier and Briggs.
Exercise. Show that (7) is equivalent to
lin 1 log x
( 1 )
x
= + - - +0 -
n
n
as n ~ +00.
To avoid all confusion, we stress the fact that, at this stage of the exposition, the formula (7) is not, in itself, either a definition or a construction
of the function log. We have only shown that, if there exists a function log
satisfying log(xy) = log x + logy and such that log(l + x) rv X when x tends
to 0, then it is given by the relation (7). But we have as yet proved neither
the existence of the function log, nor the convergence of the sequence (7).
This will be the object of Theorem 3 of nO 10.
In fact, as we shall see later, 10g(1 + u) lies, for 0 < u < 1, between
u - u 2 /2 and u. Since
log x = n.log(l + x n ),
we have
(5.8)
o < nXn - log x < nx;'/2 < x 2 /2n,
the last inequality following from (6). In other words, the error committed in
replacing log x by nXn is < x 2 /2n provided that Xn < 1. It only remains, a
modest enterprise, to perform the numerical calculations.
13 This is the crucial point in obtaining formula (7). Napier, who worked at his
tables from about 1590 to his death in 1617, and Briggs, who transformed them
into logarithms to base 10 from about 1615 on, did not argue in terms of "derivatives" for the excellent reason that these would not appear, and then in a rather
hazy way, until about twenty years later, with Fermat and Descartes, Ii propos
the calculus of tangents to a curve. But Napier imagined a point moving along
a segment of line with a speed inversely proportional to its distance x from the
origin of the segment, and the concept of "instantaneous speed" was just that of
derivative with respect to time. The reader should not believe that the concepts
and ideas which, nowadays, appear simple enough to be taught every year to
hundreds of thousands of young people of the Earth, were born fully armed from
certain brains of genius, like Athena from that of Jupiter ...