Elementary arithmetic, with brief notices of its history... by Robert Potts.

30
XIv.
1. See Arts. 9, 10, 15.. See Arts. 9, 12, 15. See Art. 9.
4. If (x logta) be substituted for x in the series for ex, the series becomes identical to the series for ax.
5. See Art. 9.
6. Let logalrm=x, 1og,,x-=y; then a- = and av-n '.' (ay)'=(ax)s. Tile otler
two equations may be deduced from loga. logo,, =logct. Art. 9. Or thus
log,,m loga,nz 1
logam. lognl = log(ti.logIm, log - logl l 1,0a oe ~ iogesm log — log1 a
7. In the verification, the first of the three equations in question 6 is required.
In these verifications, it must be remembered that alogb = bloga.
S. Since yq =zP, then qlog/y=-2 logaz: but logay.logya = and log^z.Ioyc =1.
'. l oggy. lo =log,. log,. Whence -, and q log a 1 logy a.
log, a log,ce
9. Logc,{plogaan}=la.log, g l c.logp p lo c log. alog b=1,
whence log {plog1a2} = log n. log, p. log, b. logb a.
XV.
(uc-i)' (~-1)3
1. Generally logw, = (b -1)-(- 2 - + 3 -&.
2 3
_(a,-2)~
Let '-=Ca, then logeCa=m{(a-1) — c (1)
+- (1-a-. }) (1-XU)3
and let =c ' a then log=ea= {(1-a c )- 2 +- -+&c. ( 2)
2 - 3 -3
in both which series n is any number whatever, and a is a positive integer.
In series (1), at can always be found, so that cc'- 1 shall be a very small fraction:
and since the aggregates of the second and third terms, the fourth and filth, &c., are
always negative, it follows that logea is less than z(a'- 1).
In series (2), 1 - a" is a still smaller fraction, but as all the terms of this scries
1 —
are positive, it follows that loga is greater than n(1- '").
Hence logea lies between n(a-l 1) and oi(l-a —).
2. This is decldced from the expansion of ce, by taking x= -1.
( x+-1) 1 x(x-)(x( —2)~
3. (+3l)"=1+ +
z 1 1.2 2.3 x
_rtif 3) -4 &(C.
1.2 1.2.3
1 1
= 1+1 + ---+ ---+ &c., when zxis indefinitely increased.
1.2 1.2.3
4. That is, the Napierian logarithm of any number increased by unity is always
less than the number assumed.
X.2
5. Since x is very small C= 1- +-1- 1+ &c. -1+x nearly.
1.2
and eCe'=C+X=e.eX=O i- 1+-+ 2+ &c. ( =c(l+z)- nearly.
f6i ) '-11 _ 2+&c.. 1t 2 llearly- nearly.
log~-)-lo~ e +99 99 2 99) 99 2
7. Since x is very great, as x increases, ax decreases and when x becomr s indefinitely great, x may be considered equivalent to an indefinitely great power of 2.
Also x —l=(x-l)(z-+l), x-l=(x-l) (z+l), &c., whence the expression
for loge,, may be deduced.