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 Answer4

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 calculationlog10(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 calculationx = 210 = 1024y = x100 = (210)100 = 21000Therefore, y = 21000 Checklogxy = 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 Answer3.3223

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 checkSince 80<100, log ⁡ 10 80 log 10 80 should be just under 2; 1.9030 is reasonable. Final Answer1.9030

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 Answer4

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 Verification5x+1 = 16 > 0, x+5 = 8 > 0 valid. Substitute to check equality. Final Answer3

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 calculationlog10(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 Answer20

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 calculationlog10512 = log10(29) = 9 × 0.3010 = 2.709log105 = 1 − log102 = 1 − 0.3010 = 0.6990log5512 = 2.709 / 0.6990 = 3.876 (approx.) Verification53 = 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 calculationlog10(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 calculationx = 210 = 1024y = x100 = (210)100 = 21000Therefore, y = 21000 Checklogxy = 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 VerificationSince √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

Number of digits of 2^64 using log10 2: Given log10(2) = 0.3010, find the number of decimal digits in 2^64.

Accepted Answer

Introduction / Context:The number of digits of a positive integer N equals ⌊log10 N⌋ + 1. With log10 2 given, we can compute log10(2^64) and apply this rule. Given Data / Assumptions: log10 2 = 0.3010 (approximation). N = 2^64. Concept / Approach:Use logarithm laws: log10(2^64) = 64·log10 2. Then number of digits = floor(value) + 1. Step-by-Step Solution:log10(2^64) = 64 × 0.3010 = 19.264.Number of digits = ⌊19.264⌋ + 1 = 19 + 1 = 20. Verification / Alternative check:2^10 ≈ 10^3 ⇒ 2^60 ≈ 10^18 and 2^4 = 16 ⇒ roughly 1.6 × 10^19 — a 20-digit number, consistent. Why Other Options Are Wrong:19 would undercount; 21 would overcount compared to the logarithmic computation. Common Pitfalls:Forgetting to add 1 after flooring, or rounding 19.264 up instead of taking the floor. Final Answer:20

Question 16

Logarithm manipulation – simplify expression: Evaluate (log_a x)/(log_{ab} x) − log_a b.

Accepted Answer

Introduction / Context:We simplify a compound expression using change-of-base and basic logarithm identities. The expression mixes bases a and ab. Given Data / Assumptions: a > 0, a ≠ 1, x > 0, b > 0, b ≠ 1. Expression: (log_a x)/(log_{ab} x) − log_a b. Concept / Approach:Use change of base: log_{ab} x = (log_a x)/(log_a (ab)) = (log_a x)/(1 + log_a b). Then compute the ratio and subtract log_a b. Step-by-Step Solution:log_{ab} x = (log_a x)/(1 + log_a b).(log_a x)/(log_{ab} x) = (log_a x)/[(log_a x)/(1 + log_a b)] = 1 + log_a b.Therefore, (log_a x)/(log_{ab} x) − log_a b = (1 + log_a b) − log_a b = 1. Verification / Alternative check:Pick a = 10, b = 10, x arbitrary: then ab = 100 and the same simplification gives 1. Why Other Options Are Wrong:0 would require cancellation of the leading 1; a or ab are dimensionally inconsistent with a pure number result. Common Pitfalls:Misapplying change-of-base or forgetting that log_a a = 1. Final Answer:1

Question 17

Telescoping product of logs: Evaluate log₂3 × log₃2 × log₃4 × log₄3.

Accepted Answer

Introduction / Context:Products of logarithms with swapped bases often telescope. The identity log_b a × log_a b = 1 is the key. Given Data / Assumptions: All logs are valid (bases and arguments positive and not equal to 1). Concept / Approach:Group the terms in pairs that invert bases and arguments: (log₂3 × log₃2) and (log₃4 × log₄3). Each pair equals 1. Step-by-Step Solution:log₂3 × log₃2 = 1.log₃4 × log₄3 = 1.Product = 1 × 1 = 1. Verification / Alternative check:Convert everything to natural logs: (ln 3/ln 2)(ln 2/ln 3)(ln 4/ln 3)(ln 3/ln 4) = 1. Why Other Options Are Wrong:Any value ≠ 1 would contradict the base-inversion identity. Common Pitfalls:Changing bases inconsistently or attempting unnecessary numeric approximations. Final Answer:1

Question 18

Reciprocal of a common logarithm: Given log10 2 = 0.3010, compute log₂ 10.

Accepted Answer

Introduction / Context:Logs with swapped base and argument are reciprocals when expressed in a common base. We use the change-of-base formula to invert log10 2. Given Data / Assumptions: log10 2 = 0.3010. Concept / Approach:log₂ 10 = 1 / log10 2 by change of base: log₂ 10 = (ln 10)/(ln 2) and log10 2 = (ln 2)/(ln 10). Hence they are reciprocals. Step-by-Step Solution:log₂ 10 = 1 / 0.3010 = 1000/301 (using the given decimal as the fraction 301/1000). Verification / Alternative check:Numerically, 1000/301 ≈ 3.322, consistent with 10 ≈ 2^3.322. Why Other Options Are Wrong:0.3010 is log10 2, not its reciprocal; 0.6990 relates to log10 5; 699/301 is not the reciprocal of 301/1000. Common Pitfalls:Forgetting to invert; mixing up base-change direction. Final Answer:1000 / 301

Question 19

Evaluate the expression using common-base logarithm rules: log(9/8) − log(27/32) + log(3/4). Simplify by combining as a single logarithm and compute its value.

Accepted Answer

Introduction / Context:We simplify an expression with base-10 logarithms (any common base works since the base is consistent) by using product and quotient properties to combine terms, then evaluate. Given Data / Assumptions: Expression: log(9/8) − log(27/32) + log(3/4) All logs are in the same base (common logarithms). Concept / Approach:Use log A − log B = log(A/B) and log A + log B = log(A·B). Combine the three terms into one logarithm and then simplify the numeric fraction inside. Step-by-Step Solution: Start: log(9/8) − log(27/32) + log(3/4)Combine the first two: log((9/8)/(27/32)) = log((9/8) * (32/27))Multiply by the third: log( ((9/8) * (32/27)) * (3/4) )Compute (9 * 32 * 3) / (8 * 27 * 4) = (864) / (864) = 1Therefore the expression becomes log(1) Verification / Alternative check:Note that (9/8)*(3/4) = 27/32 exactly. Hence log(9/8) + log(3/4) = log(27/32), so subtracting log(27/32) yields 0, which equals log(1). Why Other Options Are Wrong:1, 2, and 3 correspond to log values of 10, 100, 1000 respectively; the combined argument equals 1, not a power of 10 other than 10^0. “None of these” is unnecessary since 0 is available and correct. Common Pitfalls:Mixing the order of operations or incorrectly adding/subtracting logs without first converting to a single argument causes errors. Always convert sums/differences of logs to a product/quotient inside a single log before evaluation. Final Answer:0

Question 20

Simplify the logarithmic expression: log(75/16) − 2·log(5/9) + log(32/343). Express the result as a single logarithm and identify its exact simplified form.

Accepted Answer

Introduction / Context:This problem tests manipulation of logarithms using the product, quotient, and power rules. We combine all terms into a single logarithm and then simplify the numeric argument exactly. Given Data / Assumptions: Expression: log(75/16) − 2·log(5/9) + log(32/343) All logs share the same base. Concept / Approach:Use log A − 2·log B + log C = log( A·C / B^2 ). Evaluate the fraction precisely and reduce to lowest terms. Step-by-Step Solution: Start with: log(75/16) − 2·log(5/9) + log(32/343)= log( (75/16) * (32/343) / ( (5/9)^2 ) )Compute inside: (75/16) * (32/343) = (7532)/(16343) = (752)/343 = 150/343Divide by (25/81): (150/343) / (25/81) = (150/343) * (81/25) = (681)/343 = 486/343Hence the expression equals log(486/343) Verification / Alternative check:Prime-factor view: 486 = 2·3^5 and 343 = 7^3, so the exact simplest form is log( (2·3^5) / 7^3 ). No further cancellation exists, confirming the simplified form. Why Other Options Are Wrong:Values like log 2, 2 log 2, log 3, or log 5 would require the fraction to simplify to 2, 4, 3, or 5 respectively, which it does not; 486/343 ≈ 1.417 is distinct. Common Pitfalls:Missing the square on the middle term (because of the coefficient 2) or dropping factors during fraction reduction can lead to incorrect constants like log 2 or log 3. Carefully track numerators and denominators. Final Answer:log(486/343)

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