7. In the expansion of (1+x)25 the coefficients of the (2r+1)th and (r+5)th terms are equal; find r. 8. Find n when the coefficients of the 16th and 26th terms of (1+x)" are equal. 9. Find the relation between r and n in order that the coefficients of (r+3)th and (2r-3)th terms of (1+x)3n may be equal. 10. Find the coefficient of 2TM in the expansion of (x2+1). 12m 11. Find the middle term of (1+x)2o in its simplest form. 12. Find the sum of the coefficients of (x+y)16. 13. Find the sum of the coefficients of (3x+y)'. 14. Find the 7th term from the beginning and the 7th term from the end of (a+2x)". 15. Expand (a2+2a+1)3 and (x2 − 4x+2)3. Expand to 4 terms the following expressions: 1 16. (1+2)3. 19. (1+3x)2. 3 17. (1+x)*. 20. (1-2)-3. 23. (1+2x) 22. (2+x)-3. 25. The 5th term and the 10th term of (1+x) 26. The 3rd term and the 11th term of (1+2x)2. 30. 122. 31. 620. 32. 31. 33. 1÷√√99. CHAPTER XL. LOGARITHMS. 398. DEFINITION. The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number. Thus if a*= N, x is called the logarithm of N to the base ɑ. Examples. (1) Since 34=81, the logarithm of 81 to base 3 is 4. (2) Since 101=10, 102=100, 103=1000,...... the natural numbers 1, 2, 3,... are respectively the logarithms of 10, 100, 1000,...... to base 10. 399. The logarithm of N to base a is usually written log, N, so that the same meaning is expressed by the two equations a*=N; x=loga N. Example. Find the logarithm of 32 5/4 to base 2/2. by definition, (2/2)*=325/4; 1 .. (2.22)=25.25; .. 2*2*+3. 3 27 hence, by equating the indices,x= 2 5 ; ..x= 2 18 5=3.6. 400. When it is understood that a particular system of logarithms is in use, the suffix denoting the base is omitted. Thus in arithmetical calculations in which 10 is the base, we usually write log 2, log 3,...... instead of log102, log103,...... Logarithms to the base 10 are known as Common Logarithms; this system was first introduced in 1615 by Briggs, a contemporary of Napier the inventor of Logarithms. PROPERTIES OF LOGARITHMS. 401. The logarithm of 1 is 0. For a=1 for all values of a; therefore log 1=0, whatever the base may be. 402. The logarithm of the base itself is 1. For a1-a; therefore log, a=1. = 403. To find the logarithm of a product. Let MN be the product; let a be the base of the system, and suppose x=loga M, ax = M, so that so that Thus the product MN=a*xα=a¤+y; whence, by definition, log, MN=x+y =loga M+loga N. Similarly, log, MNP=logaM+loga N+logaP; and so on for any number of factors. Example. log 42 = log (2 × 3 × 7) = log 2 + log 3+ log 7. 404. To find the logarithm of a fraction. M Let be the fraction, and suppose Thus the fraction y=loga N; a2 = N. M whence, by definition, loga-x-y "N "; =loga M-loga N. 15 Example. log (24)=log=log 15 – log 7 7 =log (3 × 5) - log 7=log 3 + log 5 - log 7. then whence, by definition, that is, 405. find the logarithm of a number raised to any power, integral or fractional. Let loga (MP) be required, and suppose x=log。M, so that a≈=M; Mr=(a*)=apx; Again, = log loga (M3)=px; loga (M3)=p loga M. 1. loga (M) = log. M. r √a3 3 = log a2 — log (c5b2) in terms of loga, log b 3 3 = log a − (log c3 + log b2) = log a − 5 log c – 2 log b. 406. From the equation 10a=N, it is evident that common logarithms will not in general be integral, and that they will not always be positive. For instance, 3154>108 and <104; ... log·06-2+ a fraction. 407. DEFINITION. The integral part of a logarithm is called the characteristic, and the decimal part is called the mantissa. The characteristic of the logarithm of any number to the base 10 can be written down by inspection, as we shall now show. 408. To determine the characteristic of the logarithm of any number greater than unity. It is clear that a number with two digits in its integral part lies between 101 and 102; a number with three digits in its integral part lies between 102 and 103; and so on. ence a number with n digits in its integral part lies between 10′′-1 and 10". Let N be a number whose integral part contains n digits; then N=10(n-1)+a fraction 1; .. log N=(n-1)+a fraction. Hence the characteristic is n−1; that is, the characteristic of the logarithm of a number greater than unity is less by one than the number of digits in its integral part, and is positive. 409. To determine the characteristic of the logarithm of a decimal fraction. A decimal with one cipher immediately after the decimal point, such as '0324, being greater than 01 and less than 1, lies between 10-2 and 10-1; a number with two ciphers after the decimal point lies between 10-3 and 10-2; and so on. Hence a decimal fraction with n ciphers immediately after the decimal point lies between 10-(~+1) and 10-". Let D be a decimal beginning with n ciphers; then D=10-(n+1)+a fraction; .. log D= −(n+1)+ a fraction. Hence the characteristic is - (n+1); that is, the characteristic of the logarithm of a decimal fraction is greater by unity than the number of ciphers immediately after the decimal point and is negative. ADVANTAGES OF COMMON LOGARITHMS. 410. Common logarithms, because of the two great advantages of the base 10, are in common use. These two advantages are as follows: (1) From the results already proved it is evident that the characteristics can be written down by inspection, so that only the mantissæ have to be registered in the Tables. (2) The mantissæ are the same for the logarithms of all |