5. There are three numbers in geometrical progression, whose product is 64, and the sum of their cubes 584. What are the numbers? Ans. 2, 4, and 8. 6. The sum of three numbers in geometrical progression is 13; and the sum of the extremes being multiplied by the mean term, the product is 30. What are the numbers? Ans. 1, 3, and 9. 7. 120 dollars are divided between four persons, in such a way, that their shares may be in arithmetical progression ; but if the second and third had received 12 dollars less each, and the fourth 24 dollars more, the shares would have been in geometrical progression. Required each share. Ans. Their shares were 3, 21, 39, and 57, respectively. 8. There are three numbers in geometrical progression, whose sum is 31, and the sum of the first and last is 26. What are the numbers ? Ans. 1, 5, and 25. 9. The sum of the first and second of four numbers in geometrical progression is 15, and the sum of the third and fourth 60. What are the numbers? Ans. 5, 10, 20, 40. 10. The sum of four numbers in geometrical progression is equal to the common ratio +1, and the first term is. What are the numbers? Ans. 14. 4 16 64 EXPONENTIAL EQUATIONS. (104.) An exponential equation is one in which the unknown quantity is an exponent; as ab, 3 = 243. When the exponent is a whole number, such an equation presents no difficulty; for example, if we would find the value of x in the equation 3 243, we are only to raise 3 to such a power as will equal 243; and the exponent of this power is the value of x. We find 35 = 243, and therefore x = 5. If, however, we have 3* 54, we can only obtain the value of x by approximation, and the approximate value will be a fraction. We now proceed to solve the equation 3* = 54. If we raise 3 to the third power, we have, z is therefore greater than 3; and it is less than 4, for 3181. We must therefore take for the value of x, 3+ a 1 fraction. Let that fraction be, in which x is greater than Raising both members of this equation to the a'th power, it Here the value of x' is less than 2; for 22-4; and it is greater than one, since 21 = 2; x' therefore equals 1+ a fraction. Putting x = 1 + 21 21+”= 3, or, 2 × 1 201 we have, 1 21+ Dividing by 2, and raising both members to the x" power, we have, But (), is greater than 2, and is less; wherefore, 1 1+ we have, 1 =2, or X()x" = <; we may therefore put and then we have, 1 By substituting for x', x", x", &c., their values, we have, for the value of x, the continued fraction, Ꮖ = 3 + 1+ 1+1 1+1 2+1 2. = 3,631579. Reducing this continued fraction to a simple one by (Arith. 42), we have, x = 3 + ¦ 69 = We have, therefore, 333,631579; and putting this value of in the original equation, it becomes, 33,63157954. For another example, take the equation (10)* = 2. Here it is obvious, that a must be less than unity, or between O and unity; for (10)o = 1, and (10)1 = 10. We therefore raising both members to the power z, we have, 2 = 10. Here is greater than 3, and less than 4, since 23 we have, Raising both members to the power ', we obtain, 2 = ($)". z' is here between 3 and 4, and we may make it then, 1 (#)2+?" = 2, òr 125 × (§)2 = 2; By continuing the operation, we should find z", between 9 and 10. Calling z" 9, and neglecting the succeeding frac 1 `tion, we have the following values: These values form a continued fraction, and we shall have, 1. Find the value of x in the equation 4* = 8. Xx 2. Find the value of x in the equation 8" = 3. 3. Find the value of x in the equation 8* X = Ans. 0,38. 128. Ans. 2,3333, &c. LOGARITHMS. (105.) If in the equation ab, we give b different values, the value of a remaining the same, the value of x will vary as b varies; and reciprocally, by giving a different values, the value of b will vary as x varies; from which it follows, that a being considered greater or less than unity, we may give it such an exponent as will make it equal to any number whatever. H Let us suppose a = b; we shall also have a2 b2, a3 = b2, &c. We have ao = 1 (39); hence for all numbers between 1 and b, the exponent of a will be a fraction less than unity; for all between b and b2, it will be 1+ a fraction; for all be Moreover, since 1 tween b2 and b3, 2+ a fraction, &c. 1 &c. (39), we have, a ̄1= it will follow, that for all fractional numbers between the exponent of a will be a negative fraction, less than unity; for all between 1 and 2 a fraction, &c., these last expo nents being all negative. If we take a less than unity, the same principles will apply, with this difference only; the negative powers of a will correspond to numbers greater than unity, and the positive pow ers, to numbers less than unity. For let a = − greater than unity, we shall have, 1 1 a' being --2 = 1 1= a (106.) The power to which a constant number, a, must be raised, to make it equal to a number, is called the logarithm of that number. Thus, if we have a* =y, and a* = y', x is the logarithm of y, and z' the logarithm of y'. The logarithm of a number is denoted by writing 1. before that number: thus, 1. y signifies the logarithm of y. If we take a constant number, a, and calculate by Art. (104), the values of x, which will make a 1, 2, 3, 4, 5, 6, 7, 8, &c., and arrange in a table these values of x, opposite the given numbers, we shall have a table of logarithms. The constant number, a, is called the base of the table, and in the common tables is ten; a number on some accounts most convenient, though tables may be calculated to any other base, |