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Note 4. The same rule also, among other more difficult forms of equations, succeeds very well in what are called exponential equations, or those, which have an unknown quantity for the exponent of the power ; as in the following example.

4. To find the value of x in the exponential cquation **SIOO.

For the more easy resolution of this kind of equations, it is convenient to take the logarithms of them, and then compute the terms by means of a table of logarithms. Thus, the logarithms of the two sides of the present equation are, xx log. of x=2, the log of 100. Then by a few trials it is soon perceived, that the value of mis somewhere between the two numbers 3 and 4, and indeed nearly in the middle between them, but rather nearer the latter than the former. By taking therefore first x=3'5, and then x=3.6, and working with the logarithms, the operation will be as follows :

First, suppose *3*5.
Logarithm of 3'5

0*5440680
Then 3'5 X log. 3-5

I'904238
The true number

2'000000

Error, too little,

--095762

Second, suppose x = 3.6.
Logarithm of 3:6 = 095563025
Then 36X log. 36 = 2002689
The true nurnber

2'000000

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As '098451 : 1 :: '002689 : 0'00273 Which correction, taken from 3.60000

Leaves 3.59727=x nearly.

On trial, this is found to be very little too small.

Take therefore again x=3-59727, and next x=3.59728, and repeat the operation as follows :

First, suppose x. 3'59727.
Logarithm of 3'59727 is 0'5559731
3.59727X log. of 3-59727= 1'9999854
The true number

2:0000000

Error, too little,

O'0000146

Second, suppose x3-59728.
Logarithm of 3'59728 is 0 5559743
359728 X log. of 3-59728= 1'9999953
The true number

2'0000000

Error, too little,

OʻC000047

-O'0000146

-0'0000047

O‘0000099 difference of the errors. Then, As '0000099 : '00001 :: '0000047 : 0'00000474747

Which correction, added to 359728000000

Gives nearly the value of x = 359728474747

5. To find 'the value of x in the equation *3 f-10x+

Ans. x=1100673. 6. To find the value of x in the equation X3 —2x=5.

5x=2600.

Ans. 200455T.

7. To find the value of x in the equation x3 +2x?23x = 70.

Ans. x=51349. 8. To find the value of x in the equation m3-17** + 54% 350.

Ans. X14'95407. 9. To find the value of x in the equation x4–3x*— 75X10000.

Ans. x= 10:2615. 10. To find the value of x in the equation 2x4 -16x3 +40x*-30x=-1.

Ans. x1284724. 11. To find the value of x in the equation 45 +2x4+ 3x3 +4** +5x=54321.

Ans. x=8'414455. 12. To find the value of x in the equation ** 123456789.

Ans. x=86400268.

END OF ALGEBRA.

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1. A POINT is that, which has position, but not magnitude.

2. A line is length, without breadth or thickness,

3. A surface, or superficies, is an extension, or a figure, of two dimensions, length and breadth, but without thickness.

4. A body, or solid, is a figure of three dimensions, namely, length, breadth and thickness.

Hence surfaces are the extremities of solids ; lines the extremities of surfaces; and points the extremities of lines.

5. Lines

* A TUTOR teaches Simson's Edition of Euclid's ELE. MENTS of GEOMETRY in Harvard College,

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