237 + Interview Questions in LOGARITHMS IN GENERAL AWARENESS Page 3 InterviewSolution

101.	If a = log2412, b = log3624, c = log4836, then prove that 1 + abc = 2bc.
Answer» 1 + abc = 1 + log₂₄12. log₃₆24. log₄₈36 = 1 + log₃₆12. log₄₈36 = 1 + log₄₈12 = log₄₈48 + log₄₈12 [∵ log_ax. log_ba = log_bx] = log₄₈ (48 × 12) = log₄₈ (24 × 24) = log₄₈ (24)² = 2 log₄₈24. ...(i) Also, 2bc = 2 log₃₆24. log₄₈36 = 2 log₄₈24 ...(ii) From (i) and (ii), we have RHS = LHS.

101.

If a = log2412, b = log3624, c = log4836, then prove that 1 + abc = 2bc.

Answer»

1 + abc = 1 + log₂₄12. log₃₆24. log₄₈36 = 1 + log₃₆12. log₄₈36

= 1 + log₄₈12 = log₄₈48 + log₄₈12 [∵ log_ax. log_ba = log_bx]

= log₄₈ (48 × 12) = log₄₈ (24 × 24)

= log₄₈ (24)² = 2 log₄₈24. ...(i)

Also, 2bc = 2 log₃₆24. log₄₈36 = 2 log₄₈24 ...(ii)

From (i) and (ii), we have RHS = LHS.

Discussion

102.	The value of `("log"_(a)("log"_(b)a))/("log"_(b)("log"_(a)b))`, isA. `"log"_(b)a`B. `"log"_(a) b`C. `-"log"_(a)b`D. `-"log"_(b)a`
Answer» Correct Answer - C

Discussion

103.	`2^(x)xx3^(2x)=100` then x belongs toA. (0, 3)B. (1, 3)C. (1, 2)D. (0, 2)
Answer» Correct Answer - C

Discussion

104.	If logx (a – b) – logx (a + b) = logx (\(\frac{b}{a}\)), find \(\frac{a^2}{b^2}+\frac{b^2}{a^2}\) .
Answer» Given, log_x (a – b) – log_x (a + b) = log_x (\(\frac{b}{a}\)) ⇒ log_x \(\bigg[\frac{(a-b)}{(a+b)}\bigg]\) = log_x (\(\frac{b}{a}\)) ⇒ a(a – b) = b(a + b) ⇒ a² – ab = ab + b² ⇒ a² – b² = 2ab ⇒ a² – 2ab – b² = 0 ⇒ \(\big(\frac{a}{b}\big)^2\) - 2\(\big(\frac{a}{b}\big)\) -1 = 0 This is a quadratic equation in \(\frac{a}{b}\) and the product of the roots is –1 i.e, if \(\frac{a}{b}\) is a root, then \(\big(-\frac{a}{b}\big)\) b a is the other root. Also, sum of its roots = 2 ∴ \(\big(\frac{a}{b}\big)^2\) + \(\big(\frac{a}{b}\big)^2\) = \(\frac{a^2}{b^2}\) + \(\frac{b^2}{a^2}\) = \(\bigg[\frac{a}{b}+\big(-\frac{b}{a}\big)\bigg]^2\) + 2 = 2² + 2 = 6.

104.

If logx (a – b) – logx (a + b) = logx (\(\frac{b}{a}\)), find \(\frac{a^2}{b^2}+\frac{b^2}{a^2}\) .

Answer»

Given, log_x (a – b) – log_x (a + b) = log_x (\(\frac{b}{a}\)) ⇒ log_x \(\bigg[\frac{(a-b)}{(a+b)}\bigg]\) = log_x (\(\frac{b}{a}\))

⇒ a(a – b) = b(a + b) ⇒ a² – ab = ab + b²

⇒ a² – b² = 2ab ⇒ a² – 2ab – b² = 0 ⇒ \(\big(\frac{a}{b}\big)^2\) - 2\(\big(\frac{a}{b}\big)\) -1 = 0

This is a quadratic equation in \(\frac{a}{b}\) and the product of the roots is –1 i.e, if \(\frac{a}{b}\) is a root, then \(\big(-\frac{a}{b}\big)\) b a is the other root. Also, sum of its roots = 2

∴ \(\big(\frac{a}{b}\big)^2\) + \(\big(\frac{a}{b}\big)^2\) = \(\frac{a^2}{b^2}\) + \(\frac{b^2}{a^2}\) = \(\bigg[\frac{a}{b}+\big(-\frac{b}{a}\big)\bigg]^2\) + 2 = 2² + 2 = 6.

Discussion

105.	If `("log"a)/(3) = ("log"b)/(4) = ("log"c)/(5)`, then ca equalsA. 2bB. `b^(2)`C. 8bD. 4b
Answer» Correct Answer - B

Discussion

106.	If `"log"_(2)x + "log"_(4)x + "log"_(16)x = (21)/(4)`, then x equals toA. 8B. 4C. 2D. 16
Answer» Correct Answer - A

Discussion

107.	If `(1)/("log"_(x)10) = (2)/("log"_(a)10)-2`, then x =A. `(a)/(2)`B. `(a)/(100)`C. `(a^(2))/(10)`D. `(a^(2))/(100)`
Answer» Correct Answer - D

Discussion

108.	If `"log"_(6) (x+3)-"log"_(6)x = 2`, then x =A. `(1)/(35)`B. `(3)/(35)`C. `(2)/(35)`D. `-(3)/(35)`
Answer» Correct Answer - B

Discussion

109.	If `2^(x).9^(2x+3) = 7^(x+5)`, then x =A. `(5"log"7 +6"log"3)/("log" 162-"log"7)`B. `(5"log"7-6"log"3)/("log"162 + "log" 7)`C. `(5"log" 7-6 "log" 3)/("log"162-"log"7)`D. none of these
Answer» Correct Answer - C

Discussion

110.	If `y = 2^(1//"log"_(x)8)`, then x equal toA. yB. `y^(2)`C. `y^(3)`D. none of these
Answer» Correct Answer - C

Discussion

111.	If `"log"_(y) x = "log"_(z)y = "log"_(x)z`, thenA. `x lt y lt z`B. `x gt y ge z`C. `x lt y le z`D. `x = y =z`
Answer» Correct Answer - D

Discussion

112.	If `3^(2x+1)*4^(x-1)=36` then find the value of xA. `"log"_(36)48`B. `"log"_(48)36`C. `"log"_(24)12`D. `"log"_(12)24`
Answer» Correct Answer - A

Discussion

113.	If `"log"_(x){"log"_(4)("log"_(x)(5x^(2) +4x^(3)))} =0`, thenA. 2B. 3C. 4D. 5
Answer» Correct Answer - D

Discussion

114.	If `"log"_(4) 2 + "log"_(4) 4 + "log"_(4) 16 + "log"_(4) x = 6`, then x =A. 4B. 64C. 32D. 8
Answer» Correct Answer - C

Discussion

115.	If `x^("log"_(x)(x^(2)-4x +5)) = (x-1)`, then x =A. 1B. 2C. 4D. 5
Answer» Correct Answer - B

Discussion

116.	The value of `16^("log"4^(3))`, isA. 8B. 3C. 4D. 9
Answer» Correct Answer - D We have, `16^("log"4^(3)) = 4^(2"log" 4 )""^(3) = 4^("log"4)""^(3^(2)) = 3^(2) = 9`

Discussion

117.	If `(4.2)^(x) = (0.42)^(y) = 100, " then "(1)/(x) -(1)/(y)=`A. 1B. 2C. `(1)/(2)`D. `-1`
Answer» Correct Answer - C

Discussion

118.	The value of `log_(381)7` lies between ______.A. `(1)/(3) and (1)/(2)`B. `(1)/(4) and (1)/(3)`C. `(1)/(5) and (1)/(4)`D. `(1)/(6) and (1)/(5)`
Answer» Correct Answer - B We know that `7^(3) lt 381 lt 7^(4)` `implies log_(7) 7^(3) lt log_(7) 381 lt log_(7) 7^(4)` `implies 3 lt log_(7) 381 lt 4` `implies (1)/(4) lt (1)/(log_(7) 381) lt (1)/(3)` `(1)/(4) lt log_(381) 7 lt (1)/(3) (because log_(b) a = (10)/(log_(a)b))`.

Discussion

119.	If `"log"_(3) x + "log"_(9)x^(2) + "log"_(27)x^(3) = 9`, then x =A. 3B. 9C. 27D. none of these
Answer» Correct Answer - C We have, `"log"_(3)x + "log"_(9)x^(2) + "log"_(27)x^(3) = 9` `rArr "log"_(3) x + "log"_(3^(2)) x^(2) + "log"_(3^(3)) x^(3) = 9` `rArr "log"_(3) x + (2)/(2) "log"_(3) x + (3)/(3) "log"_(3)x =9` `rArr 3 "log"_(3) x = 9 rArr "log"_(3) x = 3 rArr x = 3^(3)=27`

Discussion

120.	If `"log"_(6) {"log"_(4)(sqrt(x+4) + sqrt(x))} =0`, then x =A. 1B. `(5)/(4)`C. `(7)/(4)`D. `(9)/(4)`
Answer» Correct Answer - D

Discussion

121.	If `"log"_(7){"log"_(5)(sqrt(x+5) + sqrt(x))}=0` then x =A. 3B. 4C. 2D. none of these
Answer» Correct Answer - B

Discussion

122.	If `"log"_(12) 27 =a, "then log"_(6) 16=`A. `(3-a)/(3+a)`B. `4((3-a)/(3+a))`C. `3((4-a)/(4+a))`D. `3((4+a)/(4-a))`
Answer» Correct Answer - B

Discussion

123.	If `"log"_(10){98 + sqrt(x^(2) -12x + 36)}=2`, then x =A. 4B. 8C. 12D. 4, 8
Answer» Correct Answer - D

Discussion

124.	If `"log"_(8)x = 25 " and log"_(2) y = 50`, then x =A. `y^(3//2)`B. 2yC. yD. `(y)/(2)`
Answer» Correct Answer - A

Discussion

125.	If ` 3 +"log"_(5)x = 2"log"_(25) y`, then x equals toA. `(y)/(125)`B. `(y)/(25)`C. `(y^(2))/(25)`D. `3-(y^(2))/(25)`
Answer» Correct Answer - A

Discussion

126.	If `"log"_("cos"x) "tan" x + "log"_("sin"x) "cot" x =0,` then x =A. `n pi + (pi)/(4), n in Z`B. `2n pi + (pi)/(4), n in Z`C. `2n pi -(3pi)/(4), n in Z`D. none of these
Answer» Correct Answer - B The given equation is meaningful, if sin x, cos x, tan x, cot x are positive and `"cos" x ne 1, "sin" x ne 1`. Now, `"log"_("cos"x) "tan" x +"log"_("sin"x)"cot" x = 0` `rArr "log"_("cos"x) (("sin"x)/("cos" x)) + "log"_("sin"x) (("cos"x)/("sin"x)) =0` `rArr "log"_("cos"x) "sin"x - "log"_("cos"x) "cos" x + "log"_("sin"x) "cos" x - "log"_("sin"x) "sin" x = 0` `rArr "log"_("cos"x) "sin" x + "log"_("sin"x) "cos" x -2 =0` `rArr "log"_("cos"x) "sin"x + (1)/("log"_("cos" x) "sin"x) -2 =0` `rArr "log"_("cos"x) "sin"x = 1` `rArr "sin" x = "cos" x rArr "tan" x = 1 rArr x=2n pi + (pi)/(4), n in Z`.

Discussion

127.	If `log_(cosx) sinx>=2` and `x in [0,3pi]` then `sinx` lies in the intervalA. `[(sqrt(5)-1)/(2), 1]`B. `(0, (sqrt(5)-1)/(2)]`C. `[0, 1//2]`D. none of these
Answer» Correct Answer - B We observe that `"log"_("cos"x)sin x` is defined for ` x in (0, pi//2) cup (2pi, 5pi//2)`. Now, `"log_("cos"x) "sin"x ge 2` `rArr "sin" x le ("cos" x)^(2) " "[because 0 lt "cos" x lt 1]` `rArr "sin"^(2) x + "sin"x -1 lt 0` `rArr ("sin" x + (1)/(2))^(2) - (5)/(4) le 0` `rArr ("sin" x + (1)/(2) - (sqrt(5))/(2))("sin" x + (1)/(2) + (sqrt(5))/(2)) le 0` `rArr "sin" x + (1)/(2) -(sqrt(5))/(2) le 0 " " [because "sin" x + (1)/(2) + (sqrt(5))/(2) gt 0]` `rArr "sin" x le (sqrt(5)-1)/(2) rArr "sin" x in (0, (sqrt(5)-1)/(2)]`

Discussion

128.	If [x] denotes the greatest integer less than or equal to x, then `["log"_(10) 6730.4]=`A. 6B. 4C. 5D. none of these
Answer» Correct Answer - D We have, `10^(3) lt 6730.40 lt 10^(4)` `rArr 3 lt "log"_(10) 6730.40 lt 4 rArr ["log"_(10) 6730.40]=3`

Discussion

129.	If `x^({(3)/(4)("log"_(3)x)^(2) + ("log"_(3)x)-(5)/(4)}) = sqrt(3)`, then x hasA. all integral valuesB. two integral values and one irrational valuesC. all irrational valuesD. two rational values and an irrational value
Answer» Correct Answer - D We have, `x^({(3)/(4)("log"_(3)x)^(2) + ("log"_(3)x)-(5)/(4)}) = sqrt(3)` `rArr (3)/(4) ("log"_(3) x)^(2) + ("log"_(3) x)- (5)/(4) = "log"_(x) sqrt(3)` `rArr (3)/(4) ("log"_(3)x)^(2) + "log"_(3) x - (5)/(4) = (1)/(2"log"_(3)x)` `rArr (3)/(2) ("log"_(3)x)^(3) + 2("log"_(3)x)^(2)- (5)/(2)("log"_(3)x)-1 = 0` `rArr 3("log"_(3)x)^(3) + 4("log"_(3)x)^(2) -5("log"_(3)x) -2 =0` `rArr ("log"_(3)x-1)(3"log"_(3)x+1)("log"_(3)x+2) =0` `rArr "log"_(3)x =1, -(1)/(3), -2 rArr x = 3, 3^(-1//3), 3^(-2)` Clearly, x takes two rational values and an irrational value.

Discussion

130.	If `"log"_(e) 2."log"_(x) 27 = "log"_(10) 8."log"_(e) 10`, then x =A. 1B. 3C. 2D. 4
Answer» Correct Answer - B

Discussion

131.	The number of values of x `in [0,npi] ,n in Z` that satisfy the equation ` log_\|sinx\|(1+cosx)=2` is
Answer» Correct Answer - A We observe that `"log"_(\|"sin"x\|) (1+"cos"x) " is defined if " x ne n pi, (2n +1) (pi)/(2), n in Z`. Now, `"log"_(\|"sin"x\|) (1+"cos"x)=2` `rArr 1+"cos" x= \|"sin"x\|^(2)` `rArr "cos"^(2)x + "cos" x = 0 rArr "cos" x (1+"cos" x) = 0` But, `"cos"^(2)x + "cos" x ne 0 " for any " x in (0, n pi) - (2n-1) (pi)/(2), n in Z`. Hence, the given equation has no solution.

Discussion

132.	The value of `sum_(r=1)^(89)log_(10)(tan((pir)/180))` is equal toA. 10B. 1C. 0D. none of these
Answer» Correct Answer - C We have, `sum_(r = 1)^(89)"log"_(10) "tan" (pi r)/(180)` `= "log"_(10){"tan" (pi)/(180) xx "tan" (2pi)/(180) xx…xx "tan" (48 pi)/(180) xx…xx "tan" (89pi)/(180)}` ` ="log"_(10) 1 =0`

Discussion

133.	If `a_(n) gt a_(n-1) gt …. gt a_(2) gt a_(1) gt 1`, then the value of `"log"_(a_(1))"log"_(a_(2)) "log"_(a_(3))…."log"_(a_(n))` is equal to
Answer» Correct Answer - B We have, `"log"_(a_(1)) "log"_(a_(2)) "log"_(a_(3))…. "log"_(a_(n-1)) "log"_(a_(n)) a_(n)^(a_(n-1)^(" "a_(n-2)^(" ".^(.^(.^(a_(1)))))))` ` = "log"_(a_(1)) "log"_(a_(2)) "log"_(a_(3))…. "log"_(a_(n-1)) a_(n-1)^(" "a_(n-2)^(" ".^(.^(.^(a_(1))))))` `= "log"_(a_(1))"log"_(a_(2))"log"_(a_(3))..... "log"_(a_(n-2)) a_(n-2)^(" "a_(n-3)^(" ".^(.^(.^(a_(1))))))` `=...= "log"_(a_(1)) a_(1) =1`

Discussion

134.	`log_(a+b)(a^(3) +b^(3))-log_(a+b) (a^(2) -ab +b^(2))`= ________A. `log_(a+b)_ (a-b)`B. 2C. a+bD. 2
Answer» Correct Answer - D Apply laws of logarithm.

Discussion

135.	`(log)_2(log)_2(log)_3(log)_3 27^3`is`0`b. `1`c. `2`d.` 3`A. 1B. 0C. 3D. 2
Answer» Correct Answer - B

Discussion

136.	If `9^("log"_3("log"_(2) x)) = "log"_(2)x - ("log"_(2)x)^(2) + 1,` then x =A. 1B. `1//2`C. 3D. none of these
Answer» Correct Answer - D We have, `9^("log"3("log"_(2) x)) = "log"_(2)x - ("log"_(2)x)^(2) + 1` `rArr 3^(2"log"_(3) ("log"_(2)x)) = "log"_(2) x - ("log"_(2)x)^(2) +1` `rArr 3^("log"_(3) ("log"_(2) x)^(2)) = "log"_(2) x - ("log"_(2)x)^(2) + 1` `rArr ("log"_(2) x)^(2) = "log"_(2)x - ("log"_(2)x)^(2) + 1` `rArr 2("log"_(2) x)^(2) - "log"_(2) x - 1 =0` `rArr (2"log"_(2) x +1) ("log"_(2)x-1) = 0` `rArr "log"_(2)x = -(1)/(2), "log"_(2) x =1 rArr x = 2^(-1//2), 2`

Discussion

137.	If `n=1999!` then `sum_(x=1)^(1999) log_n x=`A. 1B. 0C. `root(1999)(1999)`D. `-1`
Answer» Correct Answer - A We have, `sum_(x =1)^(1999) "log"_(n) x` ` ="log"_(n) 1 + "log"_(n)2 + "log"_(n) 3 +. … + "log"_(n) 1999` `="log"_(n) (123……..1999) = "log"_(n) 1999! = "log"_(n) n = 1`

Discussion

138.	`16^(log_(16)25)`= ______A. 25B. 5C. 16D. 4
Answer» Correct Answer - B (i) Recall the laws of logarithm. (ii) use `alog_(a) N = N` after simplifying the given terms.

Discussion

139.	Let `a=(log)_3(log)_3 2.`An integer `k`satisfying `1
Answer» Correct Answer - A We have, `a = "log"_(3)("log"_(3) 2) rArr 3^(a) = "log"_(3) 2 rArr "log"_(2) 3 = 3^(-a)` Now, `1 lt 2^((-k+3^(-a))) lt 2` `rArr 0 lt -k + 3^(-a) lt 1` `rArr 0 lt -k + "log"_(2) 3 lt 1` `rArr k lt "log"_(2) 3 lt k +1` Now, `"log"_(2) 3 gt 1 rArr k +1 gt 1 rArr k gt0` and, `"log"_(2) 3 lt 2 rArr k lt 2` `therefore 0 lt k lt 2 rArr k = 1 " " [therefore "k is an integer"]`

Discussion

140.	`1/(log_(xy)x + log_(xy)y` = _______ ?A. 1B. 2C. 0D. `1/(log_(xy)x xx log_(xy)y)`
Answer» Correct Answer - D (i) Simplify and then use laws of logarithm. (ii) Take LCM and simplify. (iii) Use `log_(a)x+log_(a)y= log_(a)xy`

Discussion

141.	Can we write `log_(x) a/b as (log_(x)a)/(log_(x)b)` ?
Answer» Correct Answer - NO

Discussion

142.	Expand `log_(3)((xy^(2))/(z^(3)))`
Answer» Correct Answer - `log_(3)x + 2log_(3)y-3log_(x)z`

Discussion

143.	`log_(x)A^(n)` = ______
Answer» Correct Answer - `nlog_(x)A`

Discussion

144.	If `x^(3) -y^(2)= 3xy (x-y)` then `log(x-y)^(3)`= ______
Answer» Correct Answer - C (i) Apply log on bot the sides (ii) `x^(3) -y^(3) = 3xy (x-y) Rightarrow (x-y)^(3)=0`

Discussion

145.	If `x^(2) +y^(2) = 3xy` then choose the correct answer of 2log ( x - y) form the following option :A. logx- logyB. logx + logyC. log(xy)D. both(b) and ( c)
Answer» Correct Answer - B

Discussion

146.	`log_(z^(2)( x^(2)y^(2))`=?A. 2(logx + logy - logz)B. logx +logy - logzC. ` (log x^(2) = log y)/(log z)`D. `(logx + log y)/(log z)`
Answer» Correct Answer - D Apply laws of logarithm.

Discussion

147.	`log(169/9)- 2log13 + 2log 3 = ?`A. 1B. 0C. `log(13/3)`D. `log(x/(yz))`
Answer» Correct Answer - B Apply laws of logarithm.

Discussion

148.	Find the value of 32 - log34
Answer» 3² – log₃⁴ = 3² log₃3 – log³4 = 3log₃ 3² – log₃⁴ = 3log₃ 9 – log₃ 4 = 3log₃(9/4) If log M = – 2, 1423 then to get its characteristic and mantissa is made positive as following: = 9/4 = 2 1/4 (∵ a^log_ax = x)

148.

Find the value of 32 - log34

Answer»

3² – log₃⁴ = 3² log₃3 – log³4

= 3log₃ 3² – log₃⁴

= 3log₃ 9 – log₃ 4

= 3log₃(9/4)

If log M = – 2, 1423 then to get its characteristic and mantissa is made positive as following:

= 9/4 = 2 1/4 (∵ a^log_ax = x)

Discussion

149.	Given antilog (2.375) =x Characteristic of log x is ______
Answer» Correct Answer - 2

Discussion

150.	Find (17)1/2 whereas log17 = 1.2304 and antilog 0.6152 = 4.123.
Answer» Let(17)^1/2 = x Taking log on both sides, log (17)^1/2 = logx ⇒ log^1/2 = logx ⇒ 1/2 log17 = log x ⇒ log x = 0.6152 Taking antilog on both sides, antilog (log x) = antilog 0.6152 ⇒ x = 4.123 Hence, (17)^1/2 = 4.123

150.

Find (17)1/2 whereas log17 = 1.2304 and antilog 0.6152 = 4.123.

Answer»

Let(17)^1/2 = x

Taking log on both sides,

log (17)^1/2 = logx

⇒ log^1/2 = logx

⇒ 1/2 log17 = log x

⇒ log x = 0.6152

Taking antilog on both sides,

antilog (log x) = antilog 0.6152

⇒ x = 4.123

Hence, (17)^1/2 = 4.123

Discussion

Explore topic-wise InterviewSolutions in .

If a = log2412, b = log3624, c = log4836, then prove that 1 + abc = 2bc.

The value of `("log"_(a)("log"_(b)a))/("log"_(b)("log"_(a)b))`, isA. `"log"_(b)a`B. `"log"_(a) b`C. `-"log"_(a)b`D. `-"log"_(b)a`

`2^(x)xx3^(2x)=100` then x belongs toA. (0, 3)B. (1, 3)C. (1, 2)D. (0, 2)

If logx (a – b) – logx (a + b) = logx (\(\frac{b}{a}\)), find \(\frac{a^2}{b^2}+\frac{b^2}{a^2}\) .

If `("log"a)/(3) = ("log"b)/(4) = ("log"c)/(5)`, then ca equalsA. 2bB. `b^(2)`C. 8bD. 4b

If `"log"_(2)x + "log"_(4)x + "log"_(16)x = (21)/(4)`, then x equals toA. 8B. 4C. 2D. 16

If `(1)/("log"_(x)10) = (2)/("log"_(a)10)-2`, then x =A. `(a)/(2)`B. `(a)/(100)`C. `(a^(2))/(10)`D. `(a^(2))/(100)`

If `"log"_(6) (x+3)-"log"_(6)x = 2`, then x =A. `(1)/(35)`B. `(3)/(35)`C. `(2)/(35)`D. `-(3)/(35)`

If `2^(x).9^(2x+3) = 7^(x+5)`, then x =A. `(5"log"7 +6"log"3)/("log" 162-"log"7)`B. `(5"log"7-6"log"3)/("log"162 + "log" 7)`C. `(5"log" 7-6 "log" 3)/("log"162-"log"7)`D. none of these

If `y = 2^(1//"log"_(x)8)`, then x equal toA. yB. `y^(2)`C. `y^(3)`D. none of these

If `"log"_(y) x = "log"_(z)y = "log"_(x)z`, thenA. `x lt y lt z`B. `x gt y ge z`C. `x lt y le z`D. `x = y =z`

If `3^(2x+1)*4^(x-1)=36` then find the value of xA. `"log"_(36)48`B. `"log"_(48)36`C. `"log"_(24)12`D. `"log"_(12)24`

If `"log"_(x){"log"_(4)("log"_(x)(5x^(2) +4x^(3)))} =0`, thenA. 2B. 3C. 4D. 5

If `"log"_(4) 2 + "log"_(4) 4 + "log"_(4) 16 + "log"_(4) x = 6`, then x =A. 4B. 64C. 32D. 8

If `x^("log"_(x)(x^(2)-4x +5)) = (x-1)`, then x =A. 1B. 2C. 4D. 5

The value of `16^("log"4^(3))`, isA. 8B. 3C. 4D. 9

If `(4.2)^(x) = (0.42)^(y) = 100, " then "(1)/(x) -(1)/(y)=`A. 1B. 2C. `(1)/(2)`D. `-1`

The value of `log_(381)7` lies between ______.A. `(1)/(3) and (1)/(2)`B. `(1)/(4) and (1)/(3)`C. `(1)/(5) and (1)/(4)`D. `(1)/(6) and (1)/(5)`

If `"log"_(3) x + "log"_(9)x^(2) + "log"_(27)x^(3) = 9`, then x =A. 3B. 9C. 27D. none of these

If `"log"_(6) {"log"_(4)(sqrt(x+4) + sqrt(x))} =0`, then x =A. 1B. `(5)/(4)`C. `(7)/(4)`D. `(9)/(4)`

If `"log"_(7){"log"_(5)(sqrt(x+5) + sqrt(x))}=0` then x =A. 3B. 4C. 2D. none of these

If `"log"_(12) 27 =a, "then log"_(6) 16=`A. `(3-a)/(3+a)`B. `4((3-a)/(3+a))`C. `3((4-a)/(4+a))`D. `3((4+a)/(4-a))`

If `"log"_(10){98 + sqrt(x^(2) -12x + 36)}=2`, then x =A. 4B. 8C. 12D. 4, 8

If `"log"_(8)x = 25 " and log"_(2) y = 50`, then x =A. `y^(3//2)`B. 2yC. yD. `(y)/(2)`

If ` 3 +"log"_(5)x = 2"log"_(25) y`, then x equals toA. `(y)/(125)`B. `(y)/(25)`C. `(y^(2))/(25)`D. `3-(y^(2))/(25)`

If `"log"_("cos"x) "tan" x + "log"_("sin"x) "cot" x =0,` then x =A. `n pi + (pi)/(4), n in Z`B. `2n pi + (pi)/(4), n in Z`C. `2n pi -(3pi)/(4), n in Z`D. none of these

If `log_(cosx) sinx>=2` and `x in [0,3pi]` then `sinx` lies in the intervalA. `[(sqrt(5)-1)/(2), 1]`B. `(0, (sqrt(5)-1)/(2)]`C. `[0, 1//2]`D. none of these

If [x] denotes the greatest integer less than or equal to x, then `["log"_(10) 6730.4]=`A. 6B. 4C. 5D. none of these

If `x^({(3)/(4)("log"_(3)x)^(2) + ("log"_(3)x)-(5)/(4)}) = sqrt(3)`, then x hasA. all integral valuesB. two integral values and one irrational valuesC. all irrational valuesD. two rational values and an irrational value

If `"log"_(e) 2."log"_(x) 27 = "log"_(10) 8."log"_(e) 10`, then x =A. 1B. 3C. 2D. 4

The number of values of x `in [0,npi] ,n in Z` that satisfy the equation ` log_|sinx|(1+cosx)=2` is

The value of `sum_(r=1)^(89)log_(10)(tan((pir)/180))` is equal toA. 10B. 1C. 0D. none of these

If `a_(n) gt a_(n-1) gt …. gt a_(2) gt a_(1) gt 1`, then the value of `"log"_(a_(1))"log"_(a_(2)) "log"_(a_(3))…."log"_(a_(n))` is equal to

`log_(a+b)(a^(3) +b^(3))-log_(a+b) (a^(2) -ab +b^(2))`= ________A. `log_(a+b)_ (a-b)`B. 2C. a+bD. 2

`(log)_2(log)_2(log)_3(log)_3 27^3`is`0`b. `1`c. `2`d.` 3`A. 1B. 0C. 3D. 2

If `9^("log"_3("log"_(2) x)) = "log"_(2)x - ("log"_(2)x)^(2) + 1,` then x =A. 1B. `1//2`C. 3D. none of these

If `n=1999!` then `sum_(x=1)^(1999) log_n x=`A. 1B. 0C. `root(1999)(1999)`D. `-1`

`16^(log_(16)25)`= ______A. 25B. 5C. 16D. 4

Let `a=(log)_3(log)_3 2.`An integer `k`satisfying `1

`1/(log_(xy)x + log_(xy)y` = _______ ?A. 1B. 2C. 0D. `1/(log_(xy)x xx log_(xy)y)`

Can we write `log_(x) a/b as (log_(x)a)/(log_(x)b)` ?

Expand `log_(3)((xy^(2))/(z^(3)))`

`log_(x)A^(n)` = ______

If `x^(3) -y^(2)= 3xy (x-y)` then `log(x-y)^(3)`= ______

If `x^(2) +y^(2) = 3xy` then choose the correct answer of 2log ( x - y) form the following option :A. logx- logyB. logx + logyC. log(xy)D. both(b) and ( c)

`log_(z^(2)( x^(2)y^(2))`=?A. 2(logx + logy - logz)B. logx +logy - logzC. ` (log x^(2) = log y)/(log z)`D. `(logx + log y)/(log z)`

`log(169/9)- 2log13 + 2log 3 = ?`A. 1B. 0C. `log(13/3)`D. `log(x/(yz))`

Find the value of 32 - log34

Given antilog (2.375) =x Characteristic of log x is ______

Find (17)1/2 whereas log17 = 1.2304 and antilog 0.6152 = 4.123.