Q: Evaluate log ⁡ 2 16 log 2 16.

Concept / Approach Recognize 16 = 2 4 16=2 4 . Step-by-step calculation log ⁡ 2 16 = log ⁡ 2 ( 2 4 ) = 4 log 2 16=log 2 (2 4 )=4 Common pitfalls Confusing base and argument; the base is 2 here. Final Answer 4

Q: If log 10 7 = a, evaluate log 10 (1/70).

Concept / Approach Use product/quotient rules and the facts: log1010 = 1 and log101 = 0. Step-by-step calculation log10(1/70) = log101 - log1070= 0 - [log107 + log1010]= 0 - [a + 1] = -a - 1 Common pitfalls Forgetting to include the +1 from log1010. Final Answer -a - 1

Q: If log x y = 100 and log 2 x = 10, find y.

Concept / Approach From log2x = 10 ⇒ x = 210. From logxy = 100 ⇒ y = x100. Step-by-step calculation x = 210 = 1024y = x100 = (210)100 = 21000Therefore, y = 21000 Check logxy = log21021000 = 1000/10 = 100 ✔

Question 1

If log a + log b = log(a + b), then which relation holds true?

Accepted Answer

Given data log a + log b = log(a + b) Concept / Approach Use logarithm property: log x + log y = log(xy), for positive x, y (same base). Step-by-step calculation log a + log b = log(ab)Given this equals log(a + b)Therefore, ab = a + b (since logarithm is injective on positive reals)ab = a + b Verification / Notes Example: a = 2, b = 2 ⇒ ab = 4 and a + b = 4; equality holds and logs match. Common pitfalls Assuming any base or negative values; logs require positive arguments and fixed base. Final Answer ab = a + b

Question 2

If a·x = b·y (with a, b ≠ 0), then which relation is correct?

Accepted Answer

Given data a·x = b·y, with a and b nonzero. Concept / Approach Simple proportion: if products are equal, variables are in inverse ratio to coefficients. Step-by-step calculation a·x = b·y ⇒ divide both sides by a·y: (x / y) = (b / a)Therefore, x : y = b : ax : y = b : a Verification Example: let a = 2, b = 3, choose x : y = 3 : 2 ⇒ LHS = 2·3 = 6; RHS = 3·2 = 6. Common pitfalls Writing x : y = a : b (that would imply a·x and b·y are not equal unless a = b). Final Answer x : y = b : a

Question 3

Given log₁₀2 = 0.3010, find log₂10 (correct to 4 decimal places).

Accepted Answer

Concept / Approach Use the change-of-base formula: log ⁡ 𝑏 𝑎 = log ⁡ 10 𝑎 log ⁡ 10 𝑏 log b a= log 10 b log 10 a Here, log ⁡ 2 10 = log ⁡ 10 10 log ⁡ 10 2 log 2 10= log 10 2 log 10 10 . Step-by-step calculation log ⁡ 10 10 = 1 log 10 10=1 log ⁡ 2 10 = 1 0.3010 = 3.322259... log 2 10= 0.3010 1 =3.322259...Rounded to 4 d.p. ⇒ 3.3223 Common pitfalls Inverting the fraction incorrectly (using 0.3010 0.3010 instead of its reciprocal). Forgetting that log ⁡ 10 10 = 1 log 10 10=1. Final Answer 3.3223

Question 4

Given log₁₀2 = 0.3010, evaluate log₁₀80.

Accepted Answer

Concept / Approach Factorize 80 = 8 × 10 = 2 3 × 10 80=8×10=2 3 ×10. Use log rules: log ⁡ ( 𝑎 𝑏 ) = log ⁡ 𝑎 + log ⁡ 𝑏 log(ab)=loga+logb, log ⁡ ( 2 3 ) = 3 log ⁡ 2 log(2 3 )=3log2. Step-by-step calculation log ⁡ 10 80 = log ⁡ 10 ( 2 3 ) + log ⁡ 10 10 = 3 log ⁡ 10 2 + 1 log 10 80=log 10 (2 3 )+log 10 10=3log 10 2+1 = 3 ( 0.3010 ) + 1 = 0.9030 + 1 = < 𝑚 𝑎 𝑟 𝑘 > 1.9030 < / 𝑚 𝑎 𝑟 𝑘 > =3(0.3010)+1=0.9030+1=1.9030 Quick check Since 80<100, log ⁡ 10 80 log 10 80 should be just under 2; 1.9030 is reasonable. Final Answer 1.9030

Question 5

Evaluate  log ⁡ 2 16 log 2 	​  16.

Accepted Answer

Concept / Approach Recognize 16 = 2 4 16=2 4 . Step-by-step calculation log ⁡ 2 16 = log ⁡ 2 ( 2 4 ) = < 𝑚 𝑎 𝑟 𝑘 > 4 < / 𝑚 𝑎 𝑟 𝑘 > log 2 16=log 2 (2 4 )=4 Common pitfalls Confusing base and argument; the base is 2 here. Final Answer 4

Question 6

If  log ⁡ 10 5 + log ⁡ 10 ( 5 𝑥 + 1 ) = log ⁡ 10 ( 𝑥 + 5 ) + 1 log 10 	​  5+log 10 	​  (5x+1)=log 10 	​  (x+5)+1, find  𝑥 x.

Accepted Answer

Given data & domain 5x+1 > 0, x+5 > 0 (log arguments must be positive). Concept / Approach Use log properties: log ⁡ 𝑎 + log ⁡ 𝑏 = log ⁡ ( 𝑎 𝑏 ) loga+logb=log(ab) and 1 = log ⁡ 10 10 1=log 10 10. Step-by-step calculation log ⁡ 10 [ 5 ( 5 𝑥 + 1 ) ] = log ⁡ 10 [ 10 ( 𝑥 + 5 ) ] log 10 [5(5x+1)]=log 10 [10(x+5)] ⇒ 25 𝑥 + 5 = 10 𝑥 + 50 ⇒25x+5=10x+50 15 𝑥 = 45 ⇒ 𝑥 = < 𝑚 𝑎 𝑟 𝑘 > 3 < / 𝑚 𝑎 𝑟 𝑘 > 15x=45⇒x=3 Verification 5x+1 = 16 > 0, x+5 = 8 > 0 valid. Substitute to check equality. Final Answer 3

Question 7

Given log₁₀2 = 0.30103, find the number of digits in 2⁶⁴.

Accepted Answer

Concept / Approach The number of digits of a positive integer N in base 10 is ⌊log10N⌋ + 1. Step-by-step calculation log10(264) = 64 × log102 = 64 × 0.30103 = 19.26592Digits = ⌊19.26592⌋ + 1 = 19 + 1 = 20 Common pitfalls Rounding 19.26592 before taking the floor. Always floor first, then add 1. Final Answer 20

Question 8

Given log₁₀2 = 0.3010 and log₁₀3 = 0.4771, evaluate log₅512.

Accepted Answer

Concept / Approach Use change of base: log5512 = log10512 / log105. Step-by-step calculation log10512 = log10(29) = 9 × 0.3010 = 2.709log105 = 1 − log102 = 1 − 0.3010 = 0.6990log5512 = 2.709 / 0.6990 = 3.876 (approx.) Verification 53 = 125 < 512 < 625 = 54 ⇒ answer between 3 and 4, consistent.

Question 9

If log₁₀7 = a, evaluate log₁₀(1/70).

Accepted Answer

Concept / Approach Use product/quotient rules and the facts: log1010 = 1 and log101 = 0. Step-by-step calculation log10(1/70) = log101 - log1070= 0 - [log107 + log1010]= 0 - [a + 1] = -a - 1 Common pitfalls Forgetting to include the +1 from log1010. Final Answer -a - 1

Question 10

If log_x y = 100 and log₂ x = 10, find y.

Accepted Answer

Concept / Approach From log2x = 10 ⇒ x = 210. From logxy = 100 ⇒ y = x100. Step-by-step calculation x = 210 = 1024y = x100 = (210)100 = 21000Therefore, y = 21000 Check logxy = log21021000 = 1000/10 = 100 ✔

Question 11

Evaluate log(√8) in terms of log 8 (same base).

Accepted Answer

Concept / Approach Write the root as a fractional power and use the power rule: log(ak) = k · log a. Step-by-step calculation √8 = 81/2log(√8) = log(81/2) = (1/2) · log 8 Verification Since √8 < 8, its logarithm must be smaller than log 8; multiplying by 1/2 is consistent. Final Answer (1/2) log 8

Question 12

If log base x of (9/16) equals −1/2, find x.

Accepted Answer

Given data logx(9/16) = −1/2 Concept / Approach Use the definition of logarithm: logx(A) = k implies xk = A. Work carefully with negative fractional exponents. Step-by-step calculation logx(9/16) = −1/2 ⇒ x−1/2 = 9/16x−1/2 = 1/√x, so 1/√x = 9/16√x = 16/9x = (16/9)2 = 256/81x = 256/81 Verification x−1/2 = (256/81)−1/2 = (√(256/81))−1 = (16/9)−1 = 9/16, which matches. Common pitfalls Treating −1/2 as −0.5 but then squaring the wrong quantity. Inverting 9/16 instead of the base power expression. Final Answer 256/81

Question 13

If log 27 = 1.431 (common logarithm), find log 9.

Accepted Answer

Given data log 27 = 1.431 (base 10) Concept / Approach Express 27 and 9 as powers of 3. Use log(ak) = k log a. Step-by-step calculation 27 = 33 ⇒ log 27 = 3 log 3 = 1.431log 3 = 1.431 / 3 = 0.4779 = 32 ⇒ log 9 = 2 log 3 = 2 × 0.477 = 0.954 Verification Check: log 9 + log 3 = 0.954 + 0.477 = 1.431 = log 27 which is consistent. Common pitfalls Confusing base e and base 10 logs. Dividing 1.431 by 2 directly without first finding log 3. Final Answer 0.954

Question 14

Evaluate 1/log₃ 60 + 1/log₄ 60 + 1/log₅ 60.

Accepted Answer

Given data Expression: 1/log360 + 1/log460 + 1/log560 Concept / Approach Use change-of-base: 1/logba = logab. Combine logs with the same base by converting all to base 60. Step-by-step calculation 1/log360 = log6031/log460 = log6041/log560 = log605Sum = log603 + log604 + log605 = log60(3 × 4 × 5)= log6060 = 1 Verification Since 3 × 4 × 5 equals 60 exactly, the log base 60 of 60 is 1. Common pitfalls Adding denominators directly instead of transforming with change-of-base. Using natural logs without keeping bases consistent. Final Answer 1

Question 15

If log 2 = 0.3010, then the number of digits in 2 64 is?

Accepted Answer

Required answer = [64 log10 2] + 1 = [ 64 x 0.3010 ] + 1 = 19.264 + 1 = 19 + 1 = 20

Question 16

The value of $ \frac{\log_{a}{x}}{\log_{ab}{x}} - \log_{a}{b}$ is?

Accepted Answer

∵ loga x = ( logabx) / (logaba) ∴ The given expression = [[(logabx) / (logaba)] / [( logabx )] - ( logab) = (1/logaba) - logab = logaab - logab = loga(ab/b) = logaa = 1

Question 17

The value of log 23 x log 32 x log 34 x log 43 is?

Accepted Answer

Given Exp.= log23 x log 32 x log34 x log43 = (log3 / log2) x ( log2 / log3) x (log4 / log3) x (log3 / log4) = 1

Question 18

Given that log 10 2 = 0.3010, then log 2 10 is equal to?

Accepted Answer

log2 10 = log 10 / log 2 = 1 / log 2 = 1.0000 / 0.3010 = 1000 / 301

Question 19

The value of log 9/8 - log 27/32 + log3/4 is?

Accepted Answer

Given Exp. = log [{(9/8) / (27/32)} x 3/4)] = log [(9/8) x (3/4) x (32/27)] = log 1 = 0

Question 20

The simplified form of log(75/16) -2 log(5/9) +log(32/343) is?

Accepted Answer

Given Exp. = log75/16 - 2 log5/9 + log32/343 = log [(25 x 3) / (4 x 4)] - log (25/81) + log [(16 x 2) / (81 x 3)] = log(25 x 3) - log ( 4 x 4 ) - log(25) + log81 + log(16 x 2) -log (81 x 3) = log 25 + log 3 - log 16 - log 25 + log 81 + log 16 + log 2 - log 81 - log 3 = log 2

Logarithm Questions