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If a>0 and a#1, then |, , the base must be raised ae soe, , ais x” ifa* =m,, , Properties of Logarithm, , Let m and n are, and a,b,c #1, , (i) log, a=1, (ii) log, 1=0, (iii) log, a- log, b=1, (iv) log, a = log, a- log, b, 1) log, a = 1OBE, (v) log. a jog, ¢, , Positive number such that a,b,c>0, , (vi) log, (m-n) = log, m+ log, n, m, log, () = log, m—log, »,, log, m" =n log, m, (vii) log, ,(m") = z log, m, , (viii) a!" =n, , (ix) If 0 < p<, then a> b= log, a < log, b, (x) If p>, then a>b= log, a> log, b, , (xi) aot = yBe4, y>0,¢>0, , Logarithm, , Positive number m to a given base is defined as the index x to which, tain that number ie, log,, , m =x, which is read as “The logarithm of m to the base, , Characteristic and Mantissa, , In logyy N the integral part of N is called the characteristic and, decimal part of N is called the Mantissa., (i) (a) If N >1 the characteristic of logig N is one less than, the number of digits in integral part of N., (b) If 0 <N <1, the characteristic of logig N is one greater, than the number of zeros immediately after the, , decimal point and the first significant digit and is a, negative integer., , (ii) Insert decimal point in antilog of a number, (a) When characteristic is », then insert the decimal point, after (n+ Ith digit., , (b) When characteristic is #, then insert the decimal point, such that the first significant digit is at mh place., , + @ is the base of reel logarithm (Napier logarithm), , In x= log, x, +The logye is known as Naplerian constant, logie = 04342, + Log of negative integers are not defined, log, 0 is not, defined., , + Logarithmic function is positive as well as negative, but exponential function is always positive., , + The base of a logarithm is never taken as 0, negative, number and 1., , EEE

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Exercise, , . Was, Be 1062412.b =108,5 24,6 mlog at, are M 1+ abc, , is equal to, (a) 2ac, (© 2 ab “4 2 be, 2. The value of g1'/!es,3 pia Of these, (a) 49 (o) 625 +, , a, fobat? 108,517, , logy 23 "Tog, 23 ' equal to, , (a) 0 (0) 1 ©, 4. ur 1282 ony | log z 8k, , aoe C-a q—p’ then x®.y>. is equal to, , (a) @, , (©) abe, (c) ayz, (9) None of, 5. Ths, wale of es, ¥~ loge, , = * (zx)s* 18 se (xylene lon y ig equal, , 6. bare I al <M a, the thi, ie eee of @ number to the base Va is 6., a , (a) V4} ie ©) 6 (d) 512, 6 ——+—1 _ els is 1, , logs 29% log, xyz" log,, xyz “ANA 0, , (a) 0 (b) 1, , 2 (d) None of these, , 2 6 (logy xP ~ 1, , 8, If x'ta2" + (logs sf - 10 = yer then x is equal to, , (a) 3 (o) 9 (c) 27 (¢) 81, , 9. log ,otan 1° + log ,otan 2° + ...+ log ,otan 89° is equal, to, , (a) 0 (1 © 27 (@) 81, , 10. If a* =b,b* =c,c* =a, then the value of xyz is equal, to, fa 0 (o) 1 (2 (3, , 11. If x=log, 5, y =log,; 25, then which one of the, following is correct?, (a) x<y (oe) x=y, () x>y (¢) None of these, , 12. If logy) 2 =0.30103, then log ,,50 is equal to, (a) 230103 (b) 2.69897 (c) 1.69897 (a) 0.69897, , 13. If logy, x + log, x + log, x=14, then x is equal to, , (a) 32 (b) 64, (c) 256 (d) None of these, , , , 14. If log,, 27 =a. then log, 16 is equal to, , , , 3-2 3-a, ) 3, (j@e2 7} &) ccs, 42-4 (@) None of these, aimee ab and c are in, logy, (a) AP (>) GP, {c) HP (d@) None of these, , 16. If logas(x-1) <logaoo (¥-1). then x lies in the, interval, , (a) (2, «) (b) (-2.-1), (eo) (1, 2) (@) None of these, 17. If log 93 =0.477, then the number of digits in 3" is, equal to, (a) 16 0) 19 (@ 20 (o) 21, , 18. Jog x + log x* +log x® +. + log x" Seopa ie, log x + log x? +log x’ +. SE, , 2n-t 2n-1, a =F, 3(n4 2) ain=1), a oS, 19. Iflog y, [logy (x* = 12)] >0. then x lies in the interval, (a) (0, = 4) (4) (0) (- ©. 0), , (c) (=~, -2) 2.) {@) None of these, , 20. If log, (n ~k) <log, (nk) and a >1, then n lies in, , the interval, (a) [+t] (©) &k-1), (cy) A k* 1) (d) None of these, 21. What is log,,243 equal to? INDA 201801), (a) 075 (o) 1.25, (15 3, 22. What is the value of log, x*log, y* log, z*?, (NDA 2013(1)), (a) 10 (&) 20, () 9 (6) ©, 28. If (log,x) (log, 2x) (log,, y) =log, x*, then what is, the value of y? (NDA 2012 (11)], (a) 45 (b) 9, (©) 18 () 27, 24, What is the value of 2log, 2 ~ log, 92 miki, (a) 0, (2 a 1, , "eee