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

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3. If c, s be the sums to infinity and to n terms of a decreasing
geometrical progression whose first term is a;
shewthat nlog {1 =log{ 1- }.
4. If a, b, c be the pth, qth, and rth terms respectively, both of an
arithmetical, and of a geometrical series, shew that
(1). (b- c) log a + (c-a) log b + (a- b) I og c =O.
(2). q-r r-p q___q
b log - ca log lo b-b log (a
5. If (logey)" = x:
lolo, lorshow that Y-y (loge y)X g ) _ e- - 1-log x
X (log, y)x1 -
6. If loga x, log^y, logc z be in arithmetical progression having unity
for the common difference, shew that
loge logez
G = -loY-ge1 J —log logey + logeb
7. If X (Y + z- ) = (+ x-y) ( +y-) ~
log logy log a
prove that yz. zY = z". x = xyyZ.
8. Prove loga(logaN) log^(log^N) = log,(log,) _log,,(log,a)
Vlogab /log6, /logab / log^a
XVII.
Shew the truth of each of the following series for log,:1. (X - 1)- (X2-X- 2) + I (3-X-3)- &.
2. {(x (- 1)- (xm-. 1)2 + -(- 1)3 - &c.}
1 1 1
3. -~{(1-~XI) + (l.-M)2 + (1 l)3 + &.}
4. 2 { x-1' ( + 1 f - 1-+ - + x&c. }
I- lo.r + 1+ 1 3 +, 5 \ + J J
5. | loge( +- 1) + -logo(-1)- { 2x+- 1 3 (22'- 1) +
XVIII.
1. Shew that the base of Napier's system of logarithms differs
from the sum of the first n + 1 terms of the series
1 1
1 4 1 -— 2 1.2. -&c.,
by a quantity less than 1 2.
1.2,3.4...(n -).nz2'
2. Having given log,2 = '6931472; show that log,3 =10986123.
3. Given logo4 = 1'38629; find loge5 to five places of decimals.
4. Having given loge2=0-6931, log10=2 3026; find log,5, and
logell.
5. Given loge32 = 3-46573,60 and log,5 = 1'6094379; find log102.
6. Calculate log,099 to four places of decimals, having given
logel 0 = 23026 nearly.