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If a^x = b^y = c^z = d^w, then the values of x [(1 / y) + (1 / z) + (1 / w)] is

1) loga (abc) 2) loga (bcd) 3) logb (cda) 4) logc (dab) Solution: (2) loga (bcd) ax = by = cz = dw log ax = log by x log a = y log b (x /... View Article

If log_e [a + b] / 2 = (1 / 2) [log_e a + log_e b] then

1) a = b 2) a = b / 2 3) 2a = b 4) a = b / 3 Solution: (1) a = b loge [a + b] / 2 = (1 / 2) [loge a + loge b] loge [a + b] / 2 = (1 / 2)... View Article

If y = 2^1/[log_x(8)], then x =

1) y 2) y2 3) y3 4) None of these Solution: (3) y3 y = 21/[log_x(8)] y = 2log (8^x) = 2log (2^3)x = 2(⅓)log_2 (x) = 2log 2x^(⅓) y =... View Article

If a, b, c are distinct positive numbers each being different form 1 such that (log_b a . log_c a – log_a a) + (log_a b . log_c b – log_b b) + (log_a c . log_b c – log_c c) = 0, then abc is equal to

1) 0 2) e 3) 1 4) 2 5) 3 Solution: (3) 1 (logb a . logc a - loga a) + (loga b . logc b - logb b) + (loga c . logb c - logc c) = 0 (loga b... View Article

If log_√3 5 = a and log_√3 2 = b then log_√3 300 =

1) 2(a + b) 2) 2(a + b + 1) 3) 2(a + b + 2) 4) a + b + 4 5) a + b + 1 Solution: (2) 2(a + b + 1) log√3 5 = a and log√3 2 = b log√3 300 =... View Article

7^2log₇(5) =

1) log7 35 2) 5 3) 25 4) log7 25 Solution: (3) 25 72log7(5) = 7 log7 52 = 52 = 25... View Article

If x > 0 and log₃ x + log₃ (√x) + log₃ (∜x) + ….. = 4, then x equals

1) 9 2) 81 3) 1 4) 27 Solution: (1) 9 log3 x + log3 (√x) + log3 (∜x) + ..... log3 x + (1 / 2) log3 x + (1 / 4) log3 x + (1 / 8) log3 x + (1... View Article

If a = log₂ 3, b = log₂ 5 and c = log₇ 2, then log₁₄₀ 63 in terms of a,b and c is equal to

1) [2ac + 1] / [2c + abc + 1] 2) [2ac + 1] / [2a + c + a] 3) [2ac + 1] / [2c + ab + a] 4) None of these Solution: (4) None of these log140... View Article

The least value of the expression 2 log₁₀ x – log_x (0.01), for x > 1, is

1) 10 2) 2 3) – 0.01 4) none of these Solution: (4) none of these 2 log10 x - logx (0.01) = 2 log10 x - logx (10)-2 = 2 log10 x + 2 logx... View Article

The number of solutions of log₄ (x – 1) = log₂ (x – 3)

1) 3 2) 1 3) 2 4) 0 Solution: (2) 1 log4 (x - 1) = log2 (x - 3) = log4^(½) (x - 3) log4 (x - 1) = 2 log4 (x - 3) log4 (x - 1) = log4 (x -... View Article

If log₅ log₅ log₂ x = 0, then the value of x is

1) 32 2) 125 3) 625 4) 25 Solution: (1) 32 log5 log5 log2 x = 0 log5 log5 log2 x = log5 1 log5 (log2 x) = 1 log5 (log2 x) = log5 5 log2 x =... View Article

The value of log₂ 20 . log₂ 80 – log₂ 5 . log₂ 320 is

1) 5 2) 6 3) 7 4) 8 5) 10 Solution: (4) 8 log2 20 . log2 80 - log2 5 . log2 320 = (log2 4 + log2 5) (log2 24 + log2 5) - log2 5 [log2 26 +... View Article

Both the roots of the equation (x – b) (x – c) + (x – a) (x – c) + (x – a) (x – b) = 0 are always

1) positive 2) real 3) negative 4) none of these Solution: (2) real (x - b) (x - c) + (x - a) (x - c) + (x - a) (x - b) = 0 3x2 - 2 (a + b +... View Article

The value of 7 log₂ 16 / 15 + 5 log₂ 25 / 24 + 3 log₂ 81 / 80 is

1) 1 2) log2 (105) 3) log2 (9 / 80) 4) log2 (8 / 9) Solution: (1) 1 7 log2 16 / 15 + 5 log2 25 / 24 + 3 log2 81 / 80 = 7 log2 16 - 7 log2 15... View Article

If x = log_a bc, y = log_b ca and z = log_c ab, then the value of 1 / (1 + x) + 1 / (1 + y) + 1 / (1 + z) will be

1) x + y + z 2) 1 3) ab + bc + ca 4) abc Solution: (2) 1 x = loga bc, y = logb ca and z = logc ab 1 + x = loga abc, 1 + y = logb abc, 1 + z... View Article

If 2^x . 3^{x + 4} = 7^x, then x is equal to

1) [4 loge 3] / [loge 7 - loge 6] 2) [4 loge 3] / [loge 6 - loge 7] 3) [2 logee 3] / [loge 7 - loge 6] 4) [3 loge 3] / [loge 6 - logee 7]... View Article

The approximate value of ∛28 is

1) 3.0037 2) 3.037 3) 3.0086 4) 3.37 Solution: (2) 3.037 Let x = ∛28 log x = (1 / 3) log 28 log x = (1 / 3) (1.4472) = 0.4824 x = log... View Article

The value of [log₃ 5 × log₂₅ 27 × log₄₉ 7] / [log₈₁ 3] is

1) 1 2) 6 3) 2/3 4) 3 Solution: (4) 3 [log3 5 × log25 27 × log49 7] / [log81 3] = [log3 5 × log5^2 27 × log7^2 7] / [log3^4 3] = [log3 5 ×... View Article

In a right angled triangle, the sides are a, b and c with a hypotenuse and c – b ≠ 1, c + b ≠ 1. Then, the value of (log_c+b a + log_c-b a)/(2 log_c+b a × log_c-b a) will be

1) 2 2) - 1 3) 1/2 4) 1 Solution: (4) 1 (logc+b a + logc-b a) / (2 logc+b a × logc-b a) = (1 / 2) loga c - b + (1 / 2) loga c + b = (1 / 2)... View Article

If log₃ 2, log₃ (2^x -5) and log₃ (2^x – 7/2) are in AP, then x is equal to

1) 8 2) 3 3) - 8 4) - 3 Solution: (2) 3 log3 2, log3 (2x -5) and log3 (2x - 7/2) are in AP 2 log3 (2x - 5) = log3 2 + log3 (2x - (7 / 2))... View Article