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

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iV/s->2- " fom whichl x2o 2ae2 n 20~
is 0_/ ' a= e_; from = wich x=o ande-jb x = o and -.. The equation re.
duced is c2e2x-2c+10.
XI.
1. The base e of Napier's logarithms is greater than 2, but less than 3. Let
x = 2, thus xx = 22 = 4, which is greater than e; and so a fortiori is xi greater than e
for any value of x greater than 2. In like manner xt is shown to be less than e.
2. First, if x be greater than y, and y greater than e; let x=4, y=3, then xy=43
=64, and yx 3x = 81, and x5 is less than yX, in this particular case.
Next if x be greater than y and y be less than e; let y=2, x =3, then xY=32 =9
ail?lyx=2-3 =8, and xy is greater than yx, in this case.
The results will be the same for greater numbers, subject to the given conditions.
3. There is no direct method known for the solution of exponential equations of
this form. Different methods of trial have been attempted, of which the following
has the preference.
Taking the logarithms of both members of the equation x -=100, x log x= 2. The
question is thus reduced to finding a number such that the product of the number
required and its logarithm shall be equal to 2. On trial, it will be found that x is
greater than 3 but less than 4.
If x = 3, 3 log 3 =- 1 4313639; and if x =4, 4 log 3 = — 24082400. And the arithmetic
mean between these two results being very nearly equal to 2, it follows that the true
value of x is nearer to 4 than to 3.
If x=3-5, 3-5 log 3-5 =19042386; and if x=3-6, 3-6 log 3-6=2-0026890. Hence
the true value of x lies between 3-5 and 3 6.
Since the difference is small, if the increments of the logarithmns may bc taken
proportional to the increments of the numbers (as in the proportional parts for tablks
of logarithms), a correction may be found for the error in the assumed value of x.
2-0026090
1-9042386
-0984504 increment of logarithm for '1, difference of 3'6 and 3.5.
2-0026090
2.
'0026090 increment of logarithm for y, difference of 3-6 and x the
required value.
Thus - '06890027303, and y='0027303.
Hence xa=3-6 —0027303=3-5972697.
On trial this value of x will be found too small.
A nearer approximation may be found by a similar process, taking 3-59727 and
3'59728 instead of 3-5 and 3-6, and a second correction may be found, which will
give a result correct to a greater number of decimal places.
4. The equation is indeterminate, and admits of an indefinite number of solutions.
Let y== x, then x-, = (sx), and x- I = n. If?, be successively taken equal to 2, 3,
4, &c., a series of the value of x will be found, and from y =,x, a corresponding series
of the values of y will be known. The only solution in whole numbers is that in
which x = 2, y- =4.
5. Sce the solution of question 3 above.
6. Let y=ex, in the equation aex=m, then a==m; and taking the logarithms
logy = x, ylogea = logeqn, and lo0ge = - loge(logea) + loge (logetw). lience
x =ogy= l1g(l o g)) - logo(log l(lga).
By a similar process the other three equations can be solved.