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

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7. xn = am. emnr+l.
8. Theofirat two elquations are reducible to x =9 and y = 8, which give integral
values of x and y.
By eliminating y between the second two equations, x2 log 2 -x log 6 + log 3 = 0.
From the third set of two equations xy =2 and x+-y = 3.
9. Converting the two expressions into their equivalent exponential forms.
Thus, if log5(1loo25)=x and logio25=z then log5zx: and 102 =25, also
5'=z, whence 105xO 25. Also if y=logio(log525), then 510 =25.
If 105 = 510, the expressions are equivalent.
10. The equation may be put into the form (x)3-21((xj)2+147(x~)-316=0.
Let x=-z, then the equation becomes z3-21 z2+-147z-316= 0, an ordinary equation of the third degree, which has one integral root equal to 4. The equation may
be reduced to a quadratic and the other two roots can be found. The values of xv
are 4 and (17+/ 2T0}.
XII.
1. The condition leads to the equation 2 loge a=ca (log, a+-).
2. Logr,45 = 3(- ~-1)+12r.
3(m+1)+2n
3. Logx=-log 10l{a loge 10+b} {(log 10)2-loge 10-l}.
5. The question gives the equations -=10 and (x —J)2 =(x+-y).
6. By Euc. I. 47, x-+x^2x=~6, whence can be fouud the two sides containing
the right angle.
XIII.
For questions 1, 2, 3, see Arts. 6, 7, 8 and the notes.
4. To find the characteristic of the logarithm of 300 to base 5. Let log300 =x,
then 5 ==300 and x log5=log300, from which x-3 -5.... The characteristic is
3, which also appears fiom the fact that 300 in the denary scale is equal to 2220 in
the quinary.
5. These may be found as in the preceding. Let log1212000 = x, then 12x = 12000,
and x log 12=log 12000, whence x=4 2...:the characteristic is 4, and so for the
rest.
6. As the characteristic of the logarithm of a number is always less by unity
than the number of integral digits of which that number partly or wholly consists;
this property can be employed to determine the number of digits in the high powers
of numbers.
Since logj)2cr3=631oglo2=18-96489, the characteristic being 18, the number of
digits in 263 is 19.
Similarly the number of digits in 125100 is 210; in lO01lCO is 3001; and in
e000 is 435.
7. In this example the characteristic will be negative. See Art. 7.
8. Since 1000 is the least and 9999 the greatest number of 4 digits, the values of
x determined from the equations (1 08)==1000 and (1-01)x=9999, will suggest the
limits required, x=89 and 119.
9. If x denote the number, 2x=109, and xlog2=9 whence x=29-897...which
indicates that 2 must be multiplied 30 times, and 23~= 1,065,741,824.
10. Here logN =+2+m, in being the mantissa, log, V=-log, and here x=aq,
logx
logox=qlo(ga, it follows that locgAT= logeN logaN__ 2.. The characterqlog10 q q q
istic is.
q
11. See Arts. 7, 8.