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Chapter 1: Logarithm, Short Answer Type Questions, , If log,2, , 1,, , log,625=log 16, , log,10,, , then value, , (d) none of these, , (c) 1/5, , (b), , . (a) 4, , WBSCTE 2004, 2010, , of x is:, , Answer: (b), Explanation:, , log 1o16 log,10 [:log ,16= 4log 2], or, log,5=l> r = 5, log,2.4log,5= 4log,2, , log,2 log,625, , =, , or,, , 2. If, , px)=logsin, , x, , and, , v(x) =logcosx, then e2), , WBSCTE 2007, 2009, 2015], (d) none of these, , (c) 2, , (b)1, , (a) 0, , +e2v)is:, , Answer: (b), Explanation, , 2)e2ogsin x+e2log cos, of, 3. The value, , =, , eogsin+ eog cos, , =, , Sin'x+cos, , (c)4, , (b) 6, , =1, , WBSCTE 2007, 2009, 2011], (d) none of these, , log 343 is:, , (a) 3, , x, , Answer: (b), Explanation:, log343-log(V7) =6, , 4. The value of, , log, log, log81, , WBSCTE 2008, 20101, , is:, , (d) none of these, , (c) 1/3, , (b) 1, , (a) 3, Answer: (b), , Explanation:, , log, log, log581 log, lo8, log(V3, =, , 5., , =, , log, log ,8 log, log,(2), , Theequation log,x+ log,(1+x)=0 may, , =, , be written, , as:, , (a) x+x-1=0, , (b) r+x+1=0, , (c) x+x-e=0, , (d) none of these, , Answer: (a), , =, , log,3 =1, , WBSCTE 2008, 2014]
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MATHEMATICS, , M.10, , Explanation:, , log,x+log,(1+x)=0, , or, x*+x = e, , or,log,(x(x+1)) =0, , or,x+x=1, or,x* +x-1=0, , or,log(+x)=0, 6., , Solution, , of equation 221og, log,x, ,x, , (a) x 3, , WBSCTE 2009, , 64, , (c) x= 1//3, , (b) x = 1/3, , (d) none of these, , Answer: (d), , Explanation:, , 22 logX o64r , 2log,x =192, , or,, , log,x, , =, , 96 or,, , x, , (c)1, , (b)4, , =(3)", WBSCTE 2010], , 7. The value of log of 324 to the base 32 is:, , (a)8, , =, , (d) none of these, , Answer: (b), , Explanation:, log,32)=4log ,(3/2)-4xl =4, 8., , WBSCTE 2010], , If 2logn 2 = a, then a is -, , (c)4, , (b) 3, , (a) 2, , Answer: (C), , (d) none of these, , ., , Explanation:, , 2log, 2, , 2, log,(2), , 2: 24 =, 4=aa, og,, 2, , a= 4, 9. If log , log ,81 =1find x., , WBSCTE 2011], , Answer:, log, log,81 =l = log,2, , log,81 =2, , 2log,9 =2, , 'X = 9, , 10. Find the value of log tan 1° + logtan 2°+..+log tan 89°, , WBSCTE 20121, , Solution:, , +logtan 89°, logtan1°+logtan 2°+.+logcot 2°+log cot1°, , log tan1°+log tan 2° +, =, , POLT-M
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Logarithm, = logtamT°+ logter2° +, , =0., 11,, , M.11, , +log tan 45° +, , - logten 2o, , logten1, , (Ans.), , The value, , of, , log, log, log,, , 256 is, , WBSCTE 2013], (c) 4, , (b) 2, , (a)1, , (d) 0, , Answer: (a), Explanation:, , log, log, log,, , log,, , 256=, 256=, , 12. If log, log,, , log, log, log,(2)* log, log,8 log, log,(2)' log, 3 =1, =, , log, 81 =1, then, , x=, , WBSCTE 2014], , (d) none of these, , (c)1, , (6) 3, , a) 2, , =, , =, , Answer: (a), 1, , log,log, log,81, log,(3)', or, log, log,, =, , or,, , 1, , =, , 4=1, log,log,, , or,, , log, log,2 =1, , or, , log, 2 =1, , or x= 2, :, , 13. Find the value of x if log (x-2), , +log (x-3) log, =, , 2., , WBSCTE 2015], , Answer:, , log(x-2)+ log(x-3), , =, , log2, , log(x-2)(r-3) = log2, , -2(x-3)=2, -5x +6-2 0, , -5x+4=0, , x-4)(x-1)=0, x= 4, 1, , Since log of a negative number is not, x>3, ie.,, , defined,, , so, , x-3>0, , X=4., , WBSCTE 2016], , 14. When log, 3= x, the value of log, 27 =?, , (a) x, Answer: (a), , (b) 2x, , (c)x*, , (d) none of these, , Explanation:, , log,27= log,3 =log,3=x, POLr-M
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MATHEMATICS, , 5 . It, , pcos, , M.A2, , terms of, otWBSCTE, cos in tems, P, q 2017, qsin0+r 0, then the value of psin 6+qcos6, and, g in, , -, , D.., , =, , WBSCTE 2017, , Is, , +q, , (b) typ, , (a) p+q +r, , -r, , (d) none of these, , ()p-g' +r, Answer: (b), Hint:, , (psin 0 + qcos 0) = p sin 0+q'cos 0+ 2psin Ocos, , =p-pcos0+q-q sin'0 +2pcos sin, , p +9-(pcos6+ qcos 0) p +q -r, =, , WBSCTE 2017, , 16. If log, x* = 2, then value of x is, , (b)-2, , (a)2, , (d) 4, , (c) +2, , Answer: (a), Hint: log, x* =2, or,, , 2log, * = 2, , or,, , log, x = 1, , or,, , X=2, , or,, , x=2, , 17.If log, log, log, 81=1, then x=, , (a)2, , [WBSCTE 2017], (d) none of these, , (c) 1, , (b) 3, , Answer: (a), Hint:, , log, log, log,81 =log, log, log, (3) = log, log, 4 =1, , 18. The logarithm of 729 to the base 3 is, (a) 6, Answer: (a), , Explanation:, , (b) 5, , WBSCTE 2018], (c)7, , (d) none of these, , log, 729 log, 3 =6log,3=6, , Long Answer Type Questions, , Ir log, 0g logy, l08 _logz, _ proveprove that xyz=1., 1 . If, y-z, , z-x, , WBSCTE 2005, 2009, 2009, 2012, , x-y, , POLY-M
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M.13, , Logarithmn, , solul o g l o g y _log2.=k (where k is a constant), Let, y-z: - x *-y, S o l u t i o n :, , k(y-2), logy k(z-x), logz =k(x-y), =, , =, , log.r, , Now,, , we get,, adding together,, , logx+log.y+ logz, logl, log xyz=, , +, , F{V-2), , =, , [logl, , k(z-x) + k(x-y), , =, , k(y-z+z-x+X-y), , 0], , =, , xyz(Proved), , 2.1f g = logylogz prove that r.z*".y*", Z X, , y-2, , X-y, , 1., , =, , WBSCTE 2006, 2013,, , 2018], , Solution:, Let,, , logrlogy - Og=k (where k is a constant), , y-2, , z-x, , 1), , ) = k(y-2), logx= k(y-z)or.(y+:)logx k(y-z)y+z)or,log, =, , )(z+x)or,logyt= k(z-x*).-(2), k(x*-y) 3), *), =, k(x-v)(x+y)or,, log, (x+y)logz, -v)or,, Again, logz=k(x, , Again, logy=k(z-x)or,(z +x)logy, , k(:, , =, , -x, , =, , Adding the (1), (2) and (3) equations,, , log0log y*, or,, , we, , get, , log) =k{y -2)+k(-*)+k{*-P), , logxy**.*) =k(yP, , or, logx.y.z*", , =, , log1, , -z2 +2-x, or,, , +r-y), , or,, , logxr),ye**,**), , 0, , (Proved)), , r*.y*).z*=1, , 3 . I f r=log,bc,y=log ,ca and z=log,ab showthat 1+x, , =, , +, , y+1|, , z+1, , WBSCTE 2007, 2009, 2011, 2014, 2017], , Solution:, , 'X =log,bc, , a, , =, , bc, , or,, , a*", , =, , abe, , ), , a = (abc), , Again y, , =, , log , ca », , .b =(abc), , [multiplying both side by 'a'], , b, , =, , ca, , =>b", , =, , abc, , [multiplying both side by 'b], , 2), , POLY-M
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M.15, , Logarlthm, log, , , = k(c+ a) >(c-a)lo8 y =k(c -a)(¢ +a) =k(¢ -a*), , =k(a+ b)>(a-b)log z = k(a - b)(a + b) = k(a? - b*), , log 2, , (b-c)logx k{b-c){b +c) k(63-%, (c-a)logy k(c-a)(c+a) =k(c? -a'), =, , =, , =, , (a-b)log z = k(a -b)(a +b) = k(a* -b), , +logy" + logz*" =k{b* -c* +c2 -a +a -b), , logr", , log(xy** xze-*) =k.0=0 = log1, y, , (Proved), , = 1, , 6 . Find the value of f'(0) if f)=, , 1a+l+2x, , WBSCTE 2012], , 1+x,, , Solution:, , a+1+2x, , Le-s+)=, , Or, , logy=(a+1+2x)log T, logy=(a+1+2x){log(a+x)-log(1+x)}, , Diff. both side w.r.t x, we get, or,, , 21o(a+x)-log(1+)}+ (a+1+2x)L, a+x, , 1+x, , Or,, , -2log(a+x)log(1+3}+(a+1+2x) a+x, dx, , 1+x, , x*+2z|, , or,, , -)-2, Ix=0, , or,, , +x,, , loa+)-log(1+)}-+(a+142, , a+x, , l+x, , r-2(y (oga-log0)-(a+2a, 2a|, , loga +, , 2a-), a, , a, , -|a loga +1- |, , 1+aloga -a" (Ans.), , POLY-M
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MATHEMATICS, , M, , 7. Ir log.x_logy logz prove that, y+2, , z +X, , WBSCTE 201, , x+y, , Answer:, , l0gxlo8, , Let, , y+z, , 2+X, , or,, , (z+x)logx-(y+z)logy =0, , or,, , logx- logy = logy"' - logx, , Or,, , lo, , log, , and let, Ey logz, Z+X, , x+y, , x+y)logy = (z+x)logz, or,, , logy -logz' = logz' -logy", , or,, , =, , similarly, log, , logz, , -, , logy", , =log:x-log", , (2), (3), , Adding equations (1), (2) and (3) we get,, , log, , 8.If S(x)=|, , , find the value of f'(0)., , WBSCTE 2017, , Answer:, , Refer to Question No. 6 (Similar type) of Long Answer Type Questions., , POLY-M
