237 + Interview Questions in LOGARITHMS IN GENERAL AWARENESS Page 2 InterviewSolution

51.	If `log_(10)2` = 0.3010 , then `log_(10) 2000` =
Answer» Correct Answer - `3.3010`

Discussion

52.	If ` log_(10)2= 0.3010`, then the number of digits in `16^(12)` isA. 14B. 15C. 13D. 16
Answer» Correct Answer - B Find the value of `16^(12)` by applying logarithm.

Discussion

53.	Given `log_(10)x = y` if the characteristic of y is 10, then the number of digits to the left the decimal point in x is ______
Answer» Correct Answer - 11

Discussion

54.	`log_(x)x xx log_(y)y xx log_(z)z` = ______
Answer» Correct Answer - 1

Discussion

55.	The value of x satisfying log2(3x – 2) = \(\text{log}_{\frac{1}{2}}\) x is(a) – 1 (b) − \(\frac{1}{3}\)(c) \(\frac{1}{3}\)(d) 1
Answer» (d) 1 Given, log₂(3x-2) = \(\text{log}_{\frac{1}{2}}\)x ⇒ log₂(3x-2) = log_2^-1 x ⇒ log₂(3x-2) = \(\frac{1}{-1}\) log₂ x \(\bigg[\)Using log_aⁿx = \(\frac{1}{n}\) log_a x\(\bigg]\) ⇒ log₂ (3x – 2) = (– 1) log₂ x = log₂ x^{– 1}[Using n log_a x = log_a xⁿ] ⇒ (3x – 2) = x^{– 1} = \(\frac{1}{x}\) ⇒ 3x² – 2x – 1 = 0 ⇒ (3x + 1) (x – 1) = 0 ⇒ \(-\frac{1}{3}\) x = or 1 ⇒ x = 1, since log₂ (3x – 2) is not defined when x = − \(\frac{1}{3}\).

55.

The value of x satisfying log2(3x – 2) = \(\text{log}_{\frac{1}{2}}\) x is(a) – 1 (b) − \(\frac{1}{3}\)(c) \(\frac{1}{3}\)(d) 1

Answer»

(d) 1

Given, log₂(3x-2) = \(\text{log}_{\frac{1}{2}}\)x ⇒ log₂(3x-2) = log_2^-1 x

⇒ log₂(3x-2) = \(\frac{1}{-1}\) log₂ x \(\bigg[\)Using log_aⁿx = \(\frac{1}{n}\) log_a x\(\bigg]\)

⇒ log₂ (3x – 2) = (– 1) log₂ x = log₂ x^{– 1}[Using n log_a x = log_a xⁿ]

⇒ (3x – 2) = x^{– 1} = \(\frac{1}{x}\) ⇒ 3x² – 2x – 1 = 0

⇒ (3x + 1) (x – 1) = 0 ⇒ \(-\frac{1}{3}\) x = or 1

⇒ x = 1, since log₂ (3x – 2) is not defined when x = − \(\frac{1}{3}\).

Discussion

56.	If log72 = λ, then the value of log49(28) is(a) \(\frac{1}{2}\) (2λ + 1)(b) (2λ + 1) (c) 2 (2λ + 1) (d) \(\frac{3}{2}\) (2λ + 1)
Answer» (a) \(\frac{1}{2}\) (2λ + 1) log₄₉(28) = log_7² (7 x 2²) = log_7² 7 + log_7² 2² = \(\frac{1}{2}\) log₇ 7 + \(\frac{2}{2}\) log₇ 2 \(\bigg[\)Using log_aⁿ\(x\) = \(\frac{1}{n}\) log_a x, log_aⁿ (x^m) = \(\frac{m}{n}\) log_a x\(\bigg]\) = \(\frac{1}{2}\) + λ = \(\frac{1}{2}\)(2λ + 1).

56.

If log72 = λ, then the value of log49(28) is(a) \(\frac{1}{2}\) (2λ + 1)(b) (2λ + 1) (c) 2 (2λ + 1) (d) \(\frac{3}{2}\) (2λ + 1)

Answer»

(a) \(\frac{1}{2}\) (2λ + 1)

log₄₉(28) = log_7² (7 x 2²) = log_7² 7 + log_7² 2²

= \(\frac{1}{2}\) log₇ 7 + \(\frac{2}{2}\) log₇ 2

\(\bigg[\)Using log_aⁿ\(x\) = \(\frac{1}{n}\) log_a x, log_aⁿ (x^m) = \(\frac{m}{n}\) log_a x\(\bigg]\)

= \(\frac{1}{2}\) + λ = \(\frac{1}{2}\)(2λ + 1).

Discussion

57.	\(\text{log}_{\sqrt3}\) \(\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}} =\)(a) \(\frac{31}{32}\)(b) \(\frac{15}{16}\)(c) \(\frac{7}{16}\)(d) \(\frac{15}{8}\)
Answer» (d) \(\frac{15}{8}\) Given expression = \(\text{log}_{\sqrt3}\,3^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}}\) = \(\text{log}_3{^{\frac{1}{2}}}\) \(3^{\frac{15}{16}}\) = \(\frac{15}{16}\) x 2 log₃ 3 = \(\frac{15}{8}\).

57.

\(\text{log}_{\sqrt3}\) \(\sqrt{3\sqrt{3\sqrt{3\sqrt{3}}}} =\)(a) \(\frac{31}{32}\)(b) \(\frac{15}{16}\)(c) \(\frac{7}{16}\)(d) \(\frac{15}{8}\)

Answer»

(d) \(\frac{15}{8}\)

Given expression = \(\text{log}_{\sqrt3}\,3^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}}\)

= \(\text{log}_3{^{\frac{1}{2}}}\) \(3^{\frac{15}{16}}\) = \(\frac{15}{16}\) x 2 log₃ 3 = \(\frac{15}{8}\).

Discussion

58.	If log√3 5 = a and log√3 2 = b , then log√3 300 is equal to(a) a + b + 1 (b) 2(a + b + 1) (c) 2(a + b + 2) (d) (a + b + 4)
Answer» (b) 2(a + b + 1) Hint. log_√3 300 = log_√3 \(\big[\)(√3)².10²\(\big]\) = 2 log_√3√3 + 2 log_√310 = 2 + 2 log_√3(2 x 5)

58.

If log√3 5 = a and log√3 2 = b , then log√3 300 is equal to(a) a + b + 1 (b) 2(a + b + 1) (c) 2(a + b + 2) (d) (a + b + 4)

Answer»

(b) 2(a + b + 1)

Hint.

log_√3 300 = log_√3 \(\big[\)(√3)².10²\(\big]\)

= 2 log_√3√3 + 2 log_√310 = 2 + 2 log_√3(2 x 5)

Discussion

59.	The value of `log_(x^(n))y^(m)` isA. `m/n`B. mnC. `m^(n)`D. `n^(m)`
Answer» Correct Answer - A Apply laws of logarithm.

Discussion

60.	`log_(y)x = 2` then `alog_(a) (log_(x)y)`= ______A. -2B. 4C. `1/2`D. `(-1)/4`
Answer» Correct Answer - C Apply laws of logarithm.

Discussion

61.	If `3^x = 4^(x-1)` then x can not be equal toA. `(2"log"_(3)2)/(2"log"_(3)2-1)`B. `(2)/(2-"log"_(2)3)`C. `(1)/(1-"log"_(4)3)`D. `(2"log"_(2)3)/(2"log"_(2) 3-1)`
Answer» Correct Answer - D We have, `3^(x) = 4^(x-1)` `rArr x "log"_(10) 3 = (x-1)"log"_(10)4` `rArr x = (x-1) "log"_(3)4` `rArr x = 2(x-1) "log"_(3)2` `rArr x (2"log"_(3) 2-1) = 2"log"_(3)2` `rArr x = (2"log"_(3)2)/(2"log"_(3) 2-1)` Now, `x = (2"log"_(3)2)/(2"log"_(3) 2-1)` `rArr x = (2)/(2-(1)/("log"_(3)2)) = (2)/(2-"log"_(2)3)` `rArr x = (1)/(1-(1)/(2) "log"_(2)3) = (1)/(1-"log"_(2^(2))3) = (1)/(1-"log"_(4)3)` Hence, option (a), (b) and (c) are correct and option (d) is not correct.

Discussion

62.	Simplify: i. \(\bar{3}\) .5472 - \(\bar{2}\) .8371 + 1.4581ii. 1.2489 - \(\bar{1}\) .0891 + \(\bar{2}\) .8897.
Answer» i. \(\bar{3}\) .5472 - \(\bar{2}\) .8371 + 1.4581 =(\(\bar{3}\).5472 + 1.4581)- \(\bar{2}\) .8371 = \(\bar1\).0053 - \(\bar2\) .8371 = 0.1682 ii. 1.2489 - \(\bar1\).0891 + \(\bar2\) .8897 = (1.2489 + \(\bar2\) .8897) - \(\bar1\).0891 = 0.1386 - \(\bar1\).0891 = 1.0495

62.

Simplify: i. \(\bar{3}\) .5472 - \(\bar{2}\) .8371 + 1.4581ii. 1.2489 - \(\bar{1}\) .0891 + \(\bar{2}\) .8897.

Answer»

i. \(\bar{3}\) .5472 - \(\bar{2}\) .8371 + 1.4581

=(\(\bar{3}\).5472 + 1.4581)- \(\bar{2}\) .8371

= \(\bar1\).0053 - \(\bar2\) .8371

= 0.1682

ii. 1.2489 - \(\bar1\).0891 + \(\bar2\) .8897

= (1.2489 + \(\bar2\) .8897) - \(\bar1\).0891

= 0.1386 - \(\bar1\).0891

= 1.0495

Discussion

63.	`log (a+b) + log (a-b) - log (a^(2) -b^(2))=______
Answer» Correct Answer - A (i) Use the identity log m + log n - log n= log p = log mn/p (ii) Use logx+ logy = log xy and log p - log q= log (p/q)

Discussion

64.	`log_(2)log_(2)log_(5) 125 `= _______A. 4B. 8C. 0.664D. 1
Answer» Answer is (C) 0.664 log₂ log₂ log₅ 125 = log₂ log₂ log₅ 5³ = log₂ log₂ 3 log₅ 5 (∵ log_b aⁿ = n log_b a) = log₂ log₂ 3 (∵ log₅ 5 = 1) = log₂ (log 3/log 2) (∵ log_a b = log b/log a) = \(\frac{log(\frac{log\,3}{log\,2})}{log\,2}\) \(\simeq\) 0.66444870....... Correct Answer - C Apply laws of logarithm.

64.

`log_(2)log_(2)log_(5) 125 `= _______A. 4B. 8C. 0.664D. 1

Answer»

Answer is (C) 0.664

log₂ log₂ log₅ 125

= log₂ log₂ log₅ 5³

= log₂ log₂ 3 log₅ 5 (∵ log_b aⁿ = n log_b a)

= log₂ log₂ 3 (∵ log₅ 5 = 1)

= log₂ (log 3/log 2) (∵ log_a b = log b/log a)

= \(\frac{log(\frac{log\,3}{log\,2})}{log\,2}\)

\(\simeq\) 0.66444870.......

Correct Answer - C
Apply laws of logarithm.

Discussion

65.	`log_(y(x)) xx log_(x)y` =________.A. `log_(a)y`B. `log_(x)a`C. `log_(y)a`D. `log_(a)x`
Answer» Correct Answer - B Apply laws of logarithm.

Discussion

66.	If ` log_(x)y= (log_(a)y)/(P) ` then the value of P isA. `log_(y)x`B. `log_(x)a`C. `log_(a)x`D. `log_(a)y`
Answer» Correct Answer - C (i) Recall the laws of logarithm. (ii) p = `(log a^(y))/(log x^(y))` , Use `log p^(q) = q log p`.

Discussion

67.	if `x^(3) +y^(3)=-3xy (x+y).` the `log(x+y)^(3)` = _______
Answer» Correct Answer - D Find `(x +y)^(3)`

Discussion

68.	`log(a^(3) +b^(3)) -log(a+b)-log (a^(2) -ab + b^(2))` =_______A. `a^(3) -b^(3)`B. 0C. log 1D. Both (b) and ©
Answer» Correct Answer - D (i) logx - log y - logz = log `(x/(yx))` (ii) log M - log N= `log(M/N)and long M + log N = log MN`.

Discussion

69.	`log[(root3(x^(2))xxy)/(root5(z^(2)))]`=_______A. `logx^(2//3) - log z^(2//5) + log y`B. `log x^(3//2)-logy - log z^(5//2)`C. `log x^(2//3)-log y + log z^(2//5)`D. None of these
Answer» Correct Answer - A (i) Use the identities log mn= log m + log n , `log m/n ` = log m - log n and `log x^(n) = n log x`. `((root3(x^(2)).y)/(root5(z^(2))))log(((x^(2//3).y)/(z^(2//5))))` (iii) Use `log (a/b) = log a - log b and log ab = log a + 10 log b ` simplify.

Discussion

70.	Find the value of log 6 + 2 log 5 + log 4 – log 3 – log 2.
Answer» log 6 + 2 log 5 + log 4 – log 3 – log 2 = (log 6 + log 5² + log 4) – (log 3 + log 2) = log (6 x 5² x 4) – log 3 x 2 = log((6 x 5² x 4)/(3x 2)) = log (5² x 4) = log (25 x 4) = log 100 = 2 Hence, log 6 + 2 log 5 + log 4 – log 3 – log 2 = 2

70.

Find the value of log 6 + 2 log 5 + log 4 – log 3 – log 2.

Answer»

log 6 + 2 log 5 + log 4 – log 3 – log 2

= (log 6 + log 5² + log 4) – (log 3 + log 2)

= log (6 x 5² x 4) – log 3 x 2

= log((6 x 5² x 4)/(3x 2))

= log (5² x 4)

= log (25 x 4) = log 100 = 2

Hence, log 6 + 2 log 5 + log 4 – log 3 – log 2 = 2

Discussion

71.	Prove that: log10 tan 1°. log10 tan 2°……. log10 tan 89° = 0
Answer» L.H.S. = log₁₀ tan 1°. log₁₀ tan 2°…. log₁₀ tan 45° log₁₀ tan 89° = log₁₀ tan 1°. log₁₀ tan 2°…. log₁₀ (1)…. log₁₀ tan 89° = log₁₀tan 1°- log₁₀ tan 2°…. x 0 x…. log₁₀ tan 89° = 0 = R.H.S. Hence Proved.

71.

Prove that: log10 tan 1°. log10 tan 2°……. log10 tan 89° = 0

Answer»

L.H.S.

= log₁₀ tan 1°. log₁₀ tan 2°…. log₁₀ tan 45° log₁₀ tan 89°

= log₁₀ tan 1°. log₁₀ tan 2°…. log₁₀ (1)…. log₁₀ tan 89°

= log₁₀tan 1°- log₁₀ tan 2°…. x 0 x…. log₁₀ tan 89°

= 0

= R.H.S. Hence Proved.

Discussion

72.	Number log27 is :(A) Integer(B) Rational(C) Irrational(D) Prime
Answer» Answer is (C) Irrational log₂ 7 = x 7 = 2^x Taking log on both sides, log 7 = log 2^x ⇒ x log 2 = log 7 ⇒ x = log7/log2 = (0.8451)/(0.3010) = 2.8076 = irrational number

72.

Number log27 is :(A) Integer(B) Rational(C) Irrational(D) Prime

Answer»

Answer is (C) Irrational

log₂ 7 = x

7 = 2^x

Taking log on both sides,

log 7 = log 2^x
⇒ x log 2 = log 7

⇒ x = log7/log2

= (0.8451)/(0.3010)

= 2.8076

= irrational number

Discussion

73.	log√2x = 4 then value of x will be :(A) 4√2(B) 1/4(C) 4(D) 4 x √2
Answer» Answer is (C) 4 ∵ log_a n = x ∴ a^x = n Given log_√2x= 4 Then (√2)⁴ = x ⇒ x = (√2)⁴ = (2^1/2 )⁴ = 2² = 4

73.

log√2x = 4 then value of x will be :(A) 4√2(B) 1/4(C) 4(D) 4 x √2

Answer»

Answer is (C) 4

∵ log_a n = x

∴ a^x = n

Given log_√2x= 4

Then (√2)⁴ = x

⇒ x = (√2)⁴ = (2^1/2 )⁴

= 2² = 4

Discussion

74.	logx 243 = 2.5, then value of x will be :(A) 9(B) 3(C) 1(D) 81
Answer» Answer is (A) 9 ∵ log_an = x ⇒ a^x = n Similarly log_x 243 = 2.5 ⇒ 243 = x^2.5 ⇒ x^5/2 = 243 = (3)⁵ Squaring on both sides, (x^5/2)² = {(3)⁵}² ⇒ x^5/2 = (3²)² On comparing, x = 3² = 9

74.

logx 243 = 2.5, then value of x will be :(A) 9(B) 3(C) 1(D) 81

Answer»

Answer is (A) 9

∵ log_an = x

⇒ a^x = n

Similarly log_x 243 = 2.5

⇒ 243 = x^2.5

⇒ x^5/2 = 243 = (3)⁵

Squaring on both sides,

(x^5/2)² = {(3)⁵}²

⇒ x^5/2 = (3²)²

On comparing, x = 3² = 9

Discussion

75.	The value of log (1 + 2 x 3):(A) 2 log 3(B) log 1.log 2.log 3(C) log1 + log2 + log3(D) log 7
Answer» Answer is (D) log 7 log (1 + 2 x 3) = log (1 + 6) = log 7

75.

The value of log (1 + 2 x 3):(A) 2 log 3(B) log 1.log 2.log 3(C) log1 + log2 + log3(D) log 7

Answer»

Answer is (D) log 7

log (1 + 2 x 3) = log (1 + 6) = log 7

Discussion

76.	If a < 0 then value of loga0 is :(A) – ∞(B) ∞(C) 0(D) 1
Answer» Answer is (B) ∞ log 0 = ∞ If a < 0

76.

If a < 0 then value of loga0 is :(A) – ∞(B) ∞(C) 0(D) 1

Answer»

Answer is (B) ∞

log 0 = ∞

If a < 0

Discussion

77.	The value of log (m + n) is :(A) log m + log n(B) log mn(C) log m x log n(D) none of these
Answer» Answer is (D) none of these log m + log n = log mn ≠ log (m + n) log mn = log m + log n ≠ log (m + n) log m x log n ≠ log (m + n)

77.

The value of log (m + n) is :(A) log m + log n(B) log mn(C) log m x log n(D) none of these

Answer»

Answer is (D) none of these

log m + log n = log mn ≠ log (m + n)

log mn = log m + log n ≠ log (m + n)

log m x log n ≠ log (m + n)

Discussion

78.	If a > 1 then value of loga 0 is :(A) – ∞(B) ∞(C) 0(D) 1
Answer» Answer is (A) – ∞ log_a0 = – ∞ If a > 1

78.

If a > 1 then value of loga 0 is :(A) – ∞(B) ∞(C) 0(D) 1

Answer»

Answer is (A) – ∞

log_a0 = – ∞

If a > 1

Discussion

79.	If `log(x-3)+log(x+2)=log(x^(2)+x-6)`, then the real value of x, which satisfies the above equation isA. is any value of xB. is any value of x except x = 0C. is any values of x except x = 3D. Does not exist.
Answer» Correct Answer - D log a + log b = log ab .

Discussion

80.	If `log_((sqrt(bsqrt(bsqrt(bsqrt(b)))))(sqrt(asqrt(asqrt(asqrt(asqrt(a))))))=x log_(b) a`, then x =A. `(32)/(16)`B. `(31)/(15)`C. `(31)/(30)`D. `(1)/(2)`
Answer» Correct Answer - C (i) Use, `sqrt(asqrt(asqrt(a...n " terms"))) = a^((2^(n-1//2^(n)))` and then `log_(b^(n))a^(m) = (m)/(n) log_(b)a` and simplify LHS. (ii) Compare LHS and RHS and find the value of x.

Discussion

81.	find the value of `log_(sqrt8)16`
Answer» Correct Answer - `8/3`

Discussion

82.	Find the value of 7 ` log(16/15) + 5 log (25/24) + 3 log (81/80)`.A. `"log"2`B. `"log"3`C. `"log"5`D. none of these
Answer» Correct Answer - A

Discussion

83.	Log√x y :-
Answer» \(\text{log}_{\sqrt{x}}y = \frac{log\,y}{log\,\sqrt{x}}\) \(=\frac{log\,y}{\frac{1}{2}^{\text{log x}}}\) \(=\frac{2\,log\,y}{log\,x}\)

Discussion

84.	If `log_(10)4+log_(10)m = 2`, then m = ______.
Answer» Correct Answer - 25

Discussion

85.	If `log_(xyz)x + log_(xyz)y + log_(xyz)z= log_(10)p`, then p = ______.
Answer» Correct Answer - 10

Discussion

86.	If a, b, c are positive real numbers, then `a^("log"b-"log"c) xx b^("log"c-"log"a) xx c^("log"a - "log"b)`
Answer» Correct Answer - B We have, `"log" {a^("log"b-"log"c) xx b^("log"c - "log" a) xx c^("log"a-"log"b)}` `= ("log" b-"log"c)"log" a + ("log"c-"log"a)"log"b + ("log"a-"log"b)"log"c ` = 0 `a^("log"b-"log"c) xx b^("log"c - "log" a) xx c^("log"a-"log"b)=1`

Discussion

87.	If a and b are positive real numbers other than unity, then the least value of `\|"log"_(b) a + "log"_(a) b\|`, is
Answer» Correct Answer - C We have, `\|"log"_(b) a + "log"_(a)b\|` `=\|"log"_(b)a +(1)/("log"_(b)a)\| = \|x + (1)/(x)\|, "where " x = "log"_(b)a,` We know that `x+ (1)/(x) ge 2 " for all" x gt 0 " and ", x + (1)/(x) le -2 " for all" x lt 0` `therefore \|x + (1)/(x)\| ge 2 " for all" x ne0` `therefore \|"log"_(b)a + "log"_(a)b\| ge 2` Hence, the least value of `\|"log"_(b)a + "log"_(a)b\|` is 2.

Discussion

88.	If a, b, c are positive real numbers then `(1)/("log"_(a)bc_(+1)) + (1)/("log"_(b)ca_(+1)) +(1)/("log"_(c)ab_(+1))=`
Answer» Correct Answer - B We have, `(1)/("log"_(a)bc_(+1)) + (1)/("log"_(b)ca_(+1)) +(1)/("log"_(c)ab_(+1))` `=(1)/("log"_(a)bc + "log"_(a)a) + (1)/("log"_(b)ca + "log"_(b)b) + (1)/("log"_(c)ab + "log"_(c)c)` `=(1)/("log"_(a)abc) +(1)/("log"_(b)abc) + (1)/("log"_(c)abc)` ` = "log"_(abc) a + "log"_(abc) b + "log"_(abc) c = "log"_(abc)abc = 1`

Discussion

89.	If `a = "log"2, b = "log" 3, c ="log" 7 " and " 6^(x) = 7^(x+4)` then x=A. `(4b)/(c+a-b)`B. `(4c)/(a+b-c)`C. `(4b)/(c-a-b)`D. `(4a)/(a+b-c)`
Answer» Correct Answer - B We have, `6^(x) = 7^(x+4)` `rArr x "log" 6 = (x+4)"log" 7` `rArr x = (4"log"7)/("log"6-"log"7) = (4"log"7)/("log"2 + "log"3-"log"7) = (4c)/(a+b-c)`

Discussion

90.	If`y=a^(1/(1-(log)_a x))a n dz=a^(1/(1-(log)_a y)),t h e np rov et h a tx=a^(1/(1-(log)_a z))`A. `(1)/(a^(1-"log"az))`B. `(1)/(1-"log"_(a)z)`C. `(1)/(1+"log"_(z)a)`D. `(1)/(1-"log"_(z)a)`
Answer» Correct Answer - B We have, `y= a^((1)/(1-"log"_(a)z)) " and " z= a^((1)/(1-"log"_(a)y))` `rArr "log"_(a)y= (1)/(1-"log"_(a)x) " and log"_(a) z = (1)/(1-"log"_(a)y)` `rArr "log"_(a)x= ("log"_(a)y-1)/("log"_(a)y) " and "1-"log"_(a)z = ("log"_(a)y)/("log"_(a)y-1)` `rArr "log"_(a)x= (1)/(1-"log"_(a)z)` `rArr k = (1)/(1-"log"_(a)z) " " [because x = a^(k) therefore "log"_(a)x = k]`

Discussion

91.	If x =`log_3(5)` and y =`log_17( 25)`, which one of the following is correct?A. ` x lt y`B. `x = y`C. `x gt y`D. none of these
Answer» Correct Answer - C

Discussion

92.	If pqr = 1 then find the value of `log_(rq)p+log_(rp)q+log_(pq)r`.
Answer» Correct Answer - C (i) Use `log_(a) a = 1`. (ii) Replace `rq = p^(-1), rp = q^(-1)` and `pq = r^(-1)` in the given expression. (iii) Simplify and eliminate logarithms.

Discussion

93.	If `(logx)/(a^2+a b+b^2)=(logy)/(b^2+b c+c^2)=(logz)/(c^2+c a+a^2),` then `x^(a-b)y^(b-c)z^(c-a)=`
Answer» Correct Answer - C

Discussion

94.	Statement-1: `0 lt x lt y rArr "log"_(a) x gt "log"_(a) y,` where ` a gt 1` Statement-2: `"When" a gt 1, "log"_(a) x` is an increasing function.A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True.
Answer» Correct Answer - D When `a gt 1, "log"_(a)x` is an increasing function. `therefore x lt y rArr "log"_(a) x lt "log"_(a)y`

Discussion

95.	Statement -1: `0 lt x lt y rArr "log"_(a) x gt "log"_(a) y,"where" 0 lt a lt 1` Statement-2: `"log"_(a) x` is a decreasing function when `0 lt a lt 1.`A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True.
Answer» Correct Answer - A As `"log"_(a) x` is a decreasing function for all` a in (0, 1).` `therefore x lt y rArr "log"_(a) x gt "log"_(a) y`

Discussion

96.	Statement-1: `"log"_(10)x lt "log"_(pi) x lt "log"_(e) x lt "log"_(2) x` Statement-2: `x lt y rArr "log"_(a) x gt "log"_(a) y " when " 0 lt a lt 1`A. Statement-1 is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.B. Statement-1 is True, Statement-2 is true, Statement-2 is NOT a correct explanation for Statement-1.C. Statement-1 is True, Statement-2 is False.D. Statement-1 is False, Statement-2 is True.
Answer» Correct Answer - B When `x gt 1`, we have `"log"_(x) 10 gt "log"_(x) pi gt "log"_(x) e gt "log"_(x) 2 " " [because gt pi gt e gt 2]` `rArr (1)/("log"_(10)x) gt (1)/("log"_(pi) x) gt (1)/("log"_(e) x) gt (1)/("log"_(2)x)` `rArr "log"_(10) x lt "log"_(pi) x lt "log"_(e) x lt "log"_(2) x`

Discussion

97.	Find the value of logxx + logxx3 + logxx5 + ........ + logxx(2n-1).
Answer» log_xx + log_xx³ + log_xx⁵ + log_xx⁵+ ........... + log_xx^2n-1= log_x x + 3 log_x x + ..........+ (2n - 1 ) log_x x = 1 + 3 + 5 + .......... + (2n - 1) = \(\frac{n}{2}\) [1+(2n - 1)] = n² [ Using log_x x = 1 and for A.P.S_n = \(\frac{n}{2}\) (a+l)]

97.

Find the value of logxx + logxx3 + logxx5 + ........ + logxx(2n-1).

Answer»

log_xx + log_xx³ + log_xx⁵ + log_xx⁵+ ........... + log_xx^2n-1= log_x x + 3 log_x x + ..........+ (2n - 1 ) log_x x

= 1 + 3 + 5 + .......... + (2n - 1) = \(\frac{n}{2}\) [1+(2n - 1)] = n²

[ Using log_x x = 1 and for A.P.S_n = \(\frac{n}{2}\) (a+l)]

Discussion

98.	If `x_1,x_2,x_3,...` are positive numbers in G.P then `logx_n, logx_(n+1), logx_(n+2)` are inA. A.PB. G.PC. H.PD. none of these
Answer» Correct Answer - A We have, `x_(n), x_(n+1), x_(n+2)` in G.P. `rArr x_(n+1)^(2) = x_(n)x_(n+2)` `rArr 2"log"x_(n+1) = "log"x_(n) + "log"x_(n+2)` `rArr "log"x_(n), "log"x_(n+1), "log"x_(n+2)` are in A.P.

Discussion

99.	If `log_(3) .(x^(3))/(3) - 2 log_(3) 3x^(3)=a-b log_(3)x`, then find the value of a + b.A. 6B. -6C. 0D. -3
Answer» Correct Answer - C Use log a - log b = `log.(a)/(b)` and log a + log b = log ab.

Discussion

100.	If `log_(9)[(log_(8)x)]lt0`, then x belongs to ______.A. (1, 8)B. `(-oo, 8)`C. `(8, oo)`D. (8, 1)
Answer» Correct Answer - A Given, `log_(9)(log_(8) x) lt 0` `log_(8) x lt 9^(@)` `log_(7) x lt 1` `x lt 7^(1)` There force `x in` (1, 7).

Discussion

Explore topic-wise InterviewSolutions in .

If `log_(10)2` = 0.3010 , then `log_(10) 2000` =

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if `x^(3) +y^(3)=-3xy (x+y).` the `log(x+y)^(3)` = _______

`log(a^(3) +b^(3)) -log(a+b)-log (a^(2) -ab + b^(2))` =_______A. `a^(3) -b^(3)`B. 0C. log 1D. Both (b) and ©

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If a &lt; 0 then value of loga0 is :(A) – ∞(B) ∞(C) 0(D) 1

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If a &gt; 1 then value of loga 0 is :(A) – ∞(B) ∞(C) 0(D) 1

If `log(x-3)+log(x+2)=log(x^(2)+x-6)`, then the real value of x, which satisfies the above equation isA. is any value of xB. is any value of x except x = 0C. is any values of x except x = 3D. Does not exist.

If `log_((sqrt(bsqrt(bsqrt(bsqrt(b)))))(sqrt(asqrt(asqrt(asqrt(asqrt(a))))))=x log_(b) a`, then x =A. `(32)/(16)`B. `(31)/(15)`C. `(31)/(30)`D. `(1)/(2)`

find the value of `log_(sqrt8)16`

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Log√x y :-

If `log_(10)4+log_(10)m = 2`, then m = ______.

If `log_(xyz)x + log_(xyz)y + log_(xyz)z= log_(10)p`, then p = ______.

If a, b, c are positive real numbers, then `a^("log"b-"log"c) xx b^("log"c-"log"a) xx c^("log"a - "log"b)`

If a and b are positive real numbers other than unity, then the least value of `|"log"_(b) a + "log"_(a) b|`, is

If a, b, c are positive real numbers then `(1)/("log"_(a)bc_(+1)) + (1)/("log"_(b)ca_(+1)) +(1)/("log"_(c)ab_(+1))=`

If `a = "log"2, b = "log" 3, c ="log" 7 " and " 6^(x) = 7^(x+4)` then x=A. `(4b)/(c+a-b)`B. `(4c)/(a+b-c)`C. `(4b)/(c-a-b)`D. `(4a)/(a+b-c)`

If`y=a^(1/(1-(log)_a x))a n dz=a^(1/(1-(log)_a y)),t h e np rov et h a tx=a^(1/(1-(log)_a z))`A. `(1)/(a^(1-"log"az))`B. `(1)/(1-"log"_(a)z)`C. `(1)/(1+"log"_(z)a)`D. `(1)/(1-"log"_(z)a)`

If x =`log_3(5)` and y =`log_17( 25)`, which one of the following is correct?A. ` x lt y`B. `x = y`C. `x gt y`D. none of these

If pqr = 1 then find the value of `log_(rq)p+log_(rp)q+log_(pq)r`.

If `(logx)/(a^2+a b+b^2)=(logy)/(b^2+b c+c^2)=(logz)/(c^2+c a+a^2),` then `x^(a-b)*y^(b-c)*z^(c-a)=`

Find the value of logxx + logxx3 + logxx5 + ........ + logxx(2n-1).

If `x_1,x_2,x_3,...` are positive numbers in G.P then `logx_n, logx_(n+1), logx_(n+2)` are inA. A.PB. G.PC. H.PD. none of these

If `log_(3) .(x^(3))/(3) - 2 log_(3) 3x^(3)=a-b log_(3)x`, then find the value of a + b.A. 6B. -6C. 0D. -3

If `log_(9)[(log_(8)x)]lt0`, then x belongs to ______.A. (1, 8)B. `(-oo, 8)`C. `(8, oo)`D. (8, 1)

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