To find the Logarithm of a given Number, To find the Number corresponding to a given Logarithm, 70 To determine the Modulus from the Base, To determine the Base from the Modulus, BINOMIAL THEOREM AND LOGARITHMS. CHAPTER I. POWERS AND ROOTS. 1. The student is here supposed to be acquainted with the fundamental principles of algebra; at least, with simple equations, fractions, radicals and the multiplication, division &c. of powers; but, before proceeding to the consideration of the Binomial Theorem, we shall give, in the first and second chapters, a summary view of the nature of powers and roots, and of exponents in general. 2. The power of any quantity is that quantity multiplied by itself a certain number of times. Thus, axa is a power of a which is called the second power of a, because a is taken twice as a factor; and, uxaxaxa is the fourth power of a, because a is taken four times as a factor. But instead of repeating the letter in this manner, we indicate the operation by a small figure; thus, instead of axa, we write a3, to show that a is repeated twice; and, instead of axaxaxa, we write a1. This figure is called the exponent of the power. 3. The root of any quantity is a quantity which, multiplied by itself a certain number of times, produces the first quantity. Thus, 2 is the 3d root of 8, because 2x2x2=8. That is, 8 is the 3d power of 2, and 2 is the 3d root of 8. So also, a is the 3d root of a3; or, the 2d root of a2; or, the 4th root of a*, &c. Power and root are correlative terms; am is the mth power of a, and a is the mth root of am. 3 The root of a quantity is expressed by means of the radical sign ✔and an index. Thus, the 3d root of 8 is expressed 8, the 4th root of a, a; the 5th root of x3, √x3, &c.; a signifies that 5 4 5 quantity whose 4th power is a; √x2 that quantity whose 5th power is x2, &c. Another mode of notation will be shown in the next chapter. The power or root of a compound quantity is expressed with the aid of the vinculum or parenthesis. Thus, the 4th power of a+b is expressed (a+b)*; the 3d root of x2+y2+z2, √x2+y3+z3, or 3. 3 ✔(x2+y3+z3), &c. 4. The second power is frequently called the square, the second root the square root; the third power, the cube, and the third root, the cube root. The square root is denoted by the radical sign without the index. 2 Instead of a we write simply ✔a. 5. Powers of the same quantity are multiplied together by adding their exponents. Thus, a3×a2=aaa×aa=aaaaa=a5, and 5 is the sum of 3 and 2. .... ..... In general aTM Xa"=am+n; for aTM=aaa..... repeated m times, and anaaa repeated n times, so that amxa"=aaa repeated m+n times, which is the (m+n)th power of a, or am+n. In the same way, am×a1×аo ×аa....=am+n+p+4.... for any number of factors. 6. The division of powers of the same quantity is effected by subtracting their exponents. Thus, a divided by a2, is aaaaa divided by aa; that is, =aaa-a3, and 3 is the difference of 5 and 2. In general =am-n, for am=aaa repeated m times, and anɑɑɑ.... repeated n times; so that =αaα ...... am an repeated m―n times, which is the (m-n)th power of a, or am—n. 7. The multiplication and division of powers of different quanti am an ties are merely represented. The product of a3 and b3 is ab3; the a2 quotient of a divided by b3 is Sometimes quantities to be mulb3 tiplied or divided contain powers of the same quantity; these may be multiplied or divided by the above rules, and the operation upon the rest simply denoted. Thus, the product of a3b3c_and a3df is x+y6z6 a5b3cdf; the quotient of xyz" divided by axz is and ambac " α am-p bn-ac divided by abad is 8. Involution of powers is effected by multiplying the exponents. Thus, to involve a2 to the 3d power, we are to take a three times as a factor, which is a3×a3×a3, or a2+2+2, or ao, (Art. 5,) that is, the 3d power of a3 is a with the exponent 2×3. In general the nth power of am is amn; for (am)n=am×am×am repeated n times, which gives am+m+m..... with the exponent m repeated a times; that is, with the exponent mn. Hence (am)"=amn. So also (a)=αmn; (xmn)4=xmnpq. 9. The evolution of powers is effected by dividing the exponents. Thus, to extract the square root of a we divide the exponent by 2, which gives a3. The 4th root of aoo is a5; the 5th root of y3° is yo, &c. The proof follows from what was shown in the last article, since evolution is the inverse of involution. In general the nth root mn of amn is a or am. 10. The involution and evolution of the product or quotient of powers are effected in the same manner. The 4th power of ab3 a15 the a3 a10; the 4th root of x12y4 is xy; the cube root of is is asb1o; the 5th power of ..... m6 m2 cube root of is 29 n3 m 11. Since (abc.. •)m=ambmcm.. ; and ambmcm abc...., we have the following principles : The power of the product of any number of quantities is equal to the product of their powers; and, the root of the product of any number of quantities is equal to the product of their roots. 12. The root of a quantity which is not an exact power (or if a power, one whose exponent is not a multiple of the index of the root) can only be represented. For example, the 5th root of a is repre 5 4 sented thus, ✔a; the 4th root of a3, a3, &c. Hence, when it is required to extract the root of the product or quotient of several quantities, it frequently happens that we can perform the operation in respect to a part of the quantities only, and must merely represent it in the case of the others. Thus, to extract the square root of ab, we can take the square root of a2, which is a, but the square root of b we must denote by the radical sign, thus, ✅b; so that the square root of a2b is ab. In the same way we find the square root of a1b3 to be ab3. But this last expression may be reduced to a simpler form, thus; a3✅/b3=a2✅bb, in which last form we can extract the square root b3, but must indicate that of b, so that the expression becomes a2b√b. In a similar manner we find, 4 4th root of asb11a3b»b3=a2b2✩b3. 13. When the quantities have numeral coefficients, the operations of involution and evolution may always be performed upon these, and the result prefixed to the literal part. Thus, the 3d root of 8a 32a10b C 3 is 2a; the square root of 16ab is 4aab; the 5th root of 5 b is 2a√ Sometimes it is convenient to extract the root of the numeral part, even when it is not an exact power. Thus, ✓30abc=5.477a/bc nearly. 14. Thus far we have said nothing of the signs of powers and roots. If we involve a to the 2d power, we have (-a)×(-a), which, |