Question 1

Using the identity log(10) = 1 and 10 = 2 × 5, compute log 5 given log 2 = 0.3010.

Accepted Answer

Introduction / Context: We use basic base-10 logarithm properties and the factorization 10 = 2 × 5 to compute log 5 from a given log 2 value. Given Data / Assumptions: log 2 = 0.3010 All logs are common (base 10). Concept / Approach: Since log(10) = 1 and 10 = 2 × 5, it follows that log 10 = log 2 + log 5. Solve for log 5 by subtraction. Step-by-Step Solution: log 10 = 11 = log 2 + log 5log 5 = 1 − log 2 = 1 − 0.3010 = 0.6990 Verification / Alternative check: Because 2 × 5 = 10, the result must satisfy log 2 + log 5 = 1; 0.3010 + 0.6990 = 1.0000, confirming correctness. Why Other Options Are Wrong: 0.3010 is log 2, not log 5; 0.7525 has no basis here; and it is clearly possible to compute log 5 from log 2 using the identity above. Common Pitfalls: Forgetting that log 10 equals 1 or adding instead of subtracting when solving for log 5 can cause errors. Final Answer: 0.6990

Question 2

Given log 10(2) = 0.3010 and log 10(7) = 0.8451, compute log 10(2.8). Show the decomposition and calculation steps clearly.

Accepted Answer

Introduction / Context: We evaluate a common logarithm by breaking 2.8 into factors whose logs are known and using standard log identities. Given Data / Assumptions: log 10(2) = 0.3010 log 10(7) = 0.8451 We want log 10(2.8). Concept / Approach: Write 2.8 = 28/10 = (4×7)/10, then apply log(A/B) = log A − log B and log(AB) = log A + log B. Step-by-Step Solution: log(2.8) = log(28) − log(10)log(28) = log(4×7) = log 4 + log 7 = 2·log 2 + log 7Compute: 2·0.3010 + 0.8451 = 0.6020 + 0.8451 = 1.4471Subtract log 10 = 1: 1.4471 − 1 = 0.4471 Verification / Alternative check: 2.8 ≈ 10^0.4471 (since 10^0.4471 ≈ 2.8), confirming consistency with the computed value. Why Other Options Are Wrong: 1.4471 is log 28; 2.4471 or 14.471 are orders of magnitude too large; 0.4471 is the only value matching log 2.8. Common Pitfalls: Forgetting to subtract 1 when converting from log 28 to log 2.8 (since 2.8 = 28/10) yields the common mistake 1.4471 instead of 0.4471. Final Answer: 0.4471

Question 3

If log 10(2) = 0.301, evaluate log 10(50). Show your factorization and calculation.

Accepted Answer

Introduction / Context: This question checks basic fluency with common-log rules and factorization of 50 into convenient components. Given Data / Assumptions: log 10(2) = 0.301 We seek log 10(50). Concept / Approach: Use 50 = 5 × 10 and log(AB) = log A + log B. Also use log 5 = 1 − log 2 because 10 = 2 × 5. Step-by-Step Solution: log(50) = log(5) + log(10) = log 5 + 1log 5 = 1 − log 2 = 1 − 0.301 = 0.699Therefore log(50) = 0.699 + 1 = 1.699 Verification / Alternative check: Because 50 is 5 × 10, we expect log 50 to be slightly less than log 100 = 2; indeed, 1.699 fits this expectation. Why Other Options Are Wrong: 1.301 would be log(20); 2.301 is log(200); 0.699 is log 5, not log 50. Common Pitfalls: Using log 5 = log 10 − log 2 is correct; mixing the sign or adding instead of subtracting will produce the wrong intermediate value. Final Answer: 1.699

Question 4

Evaluate: log(a^2 / (b c)) + log(b^2 / (a c)) + log(c^2 / (a b)). Simplify to a single constant value.

Accepted Answer

Introduction / Context: We simplify a sum of logarithms involving symmetric rational expressions in a, b, and c. The task is purely algebraic, requiring log combination rules. Given Data / Assumptions: Expression: log(a^2/(bc)) + log(b^2/(ac)) + log(c^2/(ab)) a, b, c > 0 (so logs are defined). Concept / Approach: Use log A + log B + log C = log(ABC). Multiply the three rational expressions and simplify the resulting fraction by collecting powers of a, b, c in numerator and denominator. Step-by-Step Solution: Product inside one log: (a^2/(bc))·(b^2/(ac))·(c^2/(ab))Numerator: a^2 b^2 c^2Denominator: (b c)(a c)(a b) = a^2 b^2 c^2Therefore the product equals 1Hence the sum equals log(1) = 0 Verification / Alternative check: By exponent rules: exponents of a, b, and c in the product are 2 − 1 − 1 = 0 each, confirming that every variable cancels and the product is 1. Why Other Options Are Wrong: 1 would equal log(10) in base 10, not log(1); abc or ab2c2 are not constants and do not reflect the cancellation evident above. Common Pitfalls: Forgetting to add exponents in the denominator or mishandling cancellations can obscure that the product equals 1 and the log is 0. Final Answer: 0

Question 5

Compute the exact value of log 8 + log(1/8) using logarithm identities.

Accepted Answer

Introduction / Context: This is a direct application of the product rule of logarithms and the defining property log(1) = 0. Given Data / Assumptions: Expression: log 8 + log(1/8) Common base for both logs (base 10 is standard). Concept / Approach: Use log A + log B = log(A·B). Multiplying 8 by its reciprocal gives 1, whose logarithm is zero in any base. Step-by-Step Solution: log 8 + log(1/8) = log(8 × 1/8) = log(1) = 0 Verification / Alternative check: Since log(1) is 0 for any allowed base, the result is base-independent as long as the base is consistent across terms. Why Other Options Are Wrong: 1 or 2 would require the product inside the log to be 10 or 100 (base 10), which it is not. log(64) represents a different expression entirely. Common Pitfalls: Occasionally students attempt to add arguments (8 + 1/8) rather than multiply; always add logs by multiplying their arguments. Final Answer: 0

Question 6

Rewrite and solve the zero-sum log equation: log_a x + log_a(1 + x) = 0. Convert to a single logarithm and find the equivalent quadratic.

Accepted Answer

Introduction / Context: We transform a sum of logs into a single log, use the definition log_a(1) = 0, and then rewrite the condition in polynomial form. Given Data / Assumptions: log_a x + log_a(1 + x) = 0 a > 0, a ≠ 1; x > 0 and x + 1 > 0 for log arguments. Concept / Approach: Convert the sum to log_a( x(1 + x) ). If the sum equals 0, the argument must be 1, because log_a(1) = 0 for any valid base a. Step-by-Step Solution: log_a x + log_a(1 + x) = log_a( x(1 + x) )Set equal to 0 ⇒ x(1 + x) = 1Expand: x^2 + x − 1 = 0 Verification / Alternative check: If x solves x^2 + x − 1 = 0 and satisfies domain constraints, then x(1 + x) = 1, so log_a( x(1 + x) ) = log_a(1) = 0, confirming equivalence. Why Other Options Are Wrong: Changing the constant term or its sign produces a different required product, not equal to 1; options with “e” are irrelevant constants here. Common Pitfalls: Forgetting the domain (x > 0) or failing to combine logs correctly can produce extraneous or invalid results. Final Answer: x2 + x - 1 = 0

Question 7

Evaluate log x + log(1/x) by combining into a single logarithm. State the exact value.

Accepted Answer

Introduction / Context: We evaluate a basic identity involving a variable and its reciprocal inside logarithms of the same base. Given Data / Assumptions: x > 0 Expression: log x + log(1/x) Concept / Approach: Use log A + log B = log(AB). Since x · (1/x) = 1, the expression reduces to log(1). Step-by-Step Solution: log x + log(1/x) = log( x · 1/x ) = log(1) = 0 Verification / Alternative check: Regardless of base (consistent across both logs), log(1) equals 0, so the value is unambiguously 0. Why Other Options Are Wrong: 1 or −1 would correspond to log(10) or log(0.1) in base 10, but the argument here equals 1; 1/2 is likewise irrelevant. Common Pitfalls: Attempting to add arguments rather than multiply (x + 1/x) is a different expression and not applicable here. Final Answer: 0

Question 8

Given log 10(90) = 1.9542, find log 10(3). Use the relationships among 3, 9, and 90.

Accepted Answer

Introduction / Context: This problem uses the factorization of 90 and standard log identities to isolate log 3. Given Data / Assumptions: log 10(90) = 1.9542 90 = 9 × 10 = 3^2 × 10 Concept / Approach: Use log(AB) = log A + log B and power rules to relate log 90 to log 3. Step-by-Step Solution: log 90 = log(3^2 × 10) = 2·log 3 + log 101.9542 = 2·log 3 + 12·log 3 = 0.9542 ⇒ log 3 = 0.9542 / 2 = 0.4771 Verification / Alternative check: Since 3 ≈ 10^0.4771, then 3^2 × 10 ≈ 10^0.9542 × 10 = 10^1.9542 ≈ 90, consistent with the given number. Why Other Options Are Wrong: Other decimals do not satisfy 2·log 3 + 1 = 1.9542 when substituted; only 0.4771 does. Common Pitfalls: Forgetting that log 10 = 1 or failing to divide by 2 after isolating 2·log 3 are common mistakes. Final Answer: 0.4771

Question 9

Solve for x in the base-4 logarithmic equation: 2·log₄(x) = 1 + log₄(x − 1). State domain conditions and the final solution.

Accepted Answer

Introduction / Context: We solve a logarithmic equation involving base 4 by converting the left-hand side with the power rule and equating arguments after combining using log identities. Given Data / Assumptions: Equation: 2·log₄(x) = 1 + log₄(x − 1) Domain: x > 0 and x − 1 > 0 ⇒ x > 1 Concept / Approach: Use 2·log₄ x = log₄(x^2) and 1 = log₄ 4. Then sum on the right: log₄ 4 + log₄(x − 1) = log₄(4(x − 1)). Equate arguments of equal logs. Step-by-Step Solution: Left: 2·log₄ x = log₄(x^2)Right: 1 + log₄(x − 1) = log₄ 4 + log₄(x − 1) = log₄(4(x − 1))Therefore x^2 = 4(x − 1)x^2 − 4x + 4 = 0 ⇒ (x − 2)^2 = 0 ⇒ x = 2 Verification / Alternative check: Check domain: x = 2 > 1; substitute: 2·log₄ 2 = 2·(1/2) = 1; right side 1 + log₄ 1 = 1 + 0 = 1. Holds exactly. Why Other Options Are Wrong: 1 violates the domain; 3 and 4 do not satisfy the quadratic; “no real solution” is incorrect since x = 2 is valid. Common Pitfalls: Forgetting to use log₄ 4 = 1 or mishandling the power rule can lead to an incorrect linear equation instead of the correct quadratic. Final Answer: 2

Question 10

Given log 10(3) = 0.477 and (1000)^x = 3, find x by applying change-of-base concepts to exponential–log relations.

Accepted Answer

Introduction / Context: We relate an exponential equation to logarithms by taking the common logarithm of both sides or by comparing powers of 10 directly. Given Data / Assumptions: (1000)^x = 3 log 10(3) = 0.477 (approximate) Concept / Approach: Since 1000 = 10^3, we have (10^3)^x = 10^{3x}. Equate exponents via logs: 10^{3x} = 3 ⇒ 3x = log 10(3) ⇒ x = log 10(3)/3. Step-by-Step Solution: Write 1000 as 10^3: (10^3)^x = 10^{3x}Set equal to 3: 10^{3x} = 3Take log base 10: 3x = log 10(3) = 0.477x = 0.477 / 3 = 0.159 Verification / Alternative check: 10^{0.477} ≈ 3, so 10^{3·0.159} ≈ 10^{0.477} ≈ 3, confirming the value of x. Why Other Options Are Wrong: Other decimals correspond to dividing by 10 or 100 instead of 3, or to unrelated operations; 10 is clearly far outside the correct scale. Common Pitfalls: Forgetting that 1000 = 10^3 and using base e without adjusting leads to numerical mistakes; the ratio approach avoids these slips. Final Answer: 0.159

Question 11

Solve for x: 5^(5 − x) = 2^(x − 5). Use logarithms (any base) to isolate x and provide the exact solution.

Accepted Answer

Introduction / Context: This is a two-base exponential equation. Taking logarithms allows moving exponents down and solving a linear equation in x. Given Data / Assumptions: 5^(5 − x) = 2^(x − 5) All quantities are positive; logs exist for any standard base. Concept / Approach: Take natural logs (or base-10 logs) of both sides: (5 − x)ln 5 = (x − 5)ln 2. Solve for x by collecting terms with x on one side and constants on the other. Step-by-Step Solution: (5 − x)ln 5 = (x − 5)ln 25ln 5 − x ln 5 = x ln 2 − 5 ln 2Bring x terms left, constants right: −x(ln 5 + ln 2) = −5(ln 2 + ln 5)Divide by −(ln 2 + ln 5): x = 5 Verification / Alternative check: Substitute x = 5: left 5^(0) = 1, right 2^(0) = 1, hence equality holds. Why Other Options Are Wrong: Values 0 or 1 do not balance both sides; “can't be determined” is incorrect as the algebra is straightforward and yields x = 5. Common Pitfalls: Dropping negative signs or forgetting that ln a + ln b = ln(ab) on both sides can produce extraneous values. Keep terms grouped carefully. Final Answer: 5

Question 12

Solve for x: log₈ x + log₄ x + log₂ x = 11. Use base conversion (let y = log₂ x) to reduce to a linear equation.

Accepted Answer

Introduction / Context: The equation mixes logarithms with bases 8, 4, and 2 applied to the same x. Converting all to base 2 simplifies the problem to a single-variable linear equation. Given Data / Assumptions: log₈ x + log₄ x + log₂ x = 11 x > 0 Concept / Approach: Let y = log₂ x, so x = 2^y. Then log₄ x = y/2 (since 4 = 2^2) and log₈ x = y/3 (since 8 = 2^3). Substitute and solve for y, then recover x = 2^y. Step-by-Step Solution: Let y = log₂ xThen log₄ x = y/2 and log₈ x = y/3Sum: y/3 + y/2 + y = 11 ⇒ (2y + 3y + 6y)/6 = 11 ⇒ 11y/6 = 11 ⇒ y = 6Therefore x = 2^6 = 64 Verification / Alternative check: Compute each term: log₂ 64 = 6, log₄ 64 = 3, log₈ 64 = 2; sum 6 + 3 + 2 = 11, confirming the solution. Why Other Options Are Wrong: 2, 4, and 8 yield sums far smaller than 11; only 64 produces the required sum. Common Pitfalls: Forgetting the base-change scaling (dividing by 2 or 3) or mis-adding fractions can produce a wrong y value. Carefully use common denominators. Final Answer: 64

Question 13

Given 10^0.3010 = 2, evaluate log base 0.125 of 125, i.e., log_{0.125}(125). Show the change-of-base steps.

Accepted Answer

Introduction / Context: We compute a logarithm with a base less than 1 using change-of-base. Given log 10(2), we can find log 10(5) and then express the result exactly as a rational combination of these constants. Given Data / Assumptions: 10^0.3010 = 2 ⇒ log 10(2) = 0.3010 We need log_{0.125}(125) Concept / Approach: Rewrite bases as powers: 0.125 = 1/8 = 2^(−3), and 125 = 5^3. Then log_{2^(−3)}(5^3) = (3 ln 5)/(−3 ln 2) = − ln 5 / ln 2. Convert to base 10 using given log values. Step-by-Step Solution: log_{0.125}(125) = ln(125)/ln(0.125) = (3 ln 5)/(ln 2^(−3)) = (3 ln 5)/(−3 ln 2) = − ln 5 / ln 2In base 10: − (log 10 5)/(log 10 2)log 10 5 = 1 − log 10 2 = 1 − 0.3010 = 0.6990Therefore value = − (0.6990)/(0.3010) = − 699/301 Verification / Alternative check: Since the base 0.125 < 1 and argument 125 > 1, the logarithm should be negative; the sign matches. The exact fraction −699/301 is the simplified ratio of common logs. Why Other Options Are Wrong: Positive 699/301 contradicts the expected sign; −1 and −2 are rough but incorrect; only −699/301 matches the exact change-of-base computation. Common Pitfalls: Forgetting that a base less than 1 flips the sign of logs relative to the same argument, or failing to compute log 5 = 1 − log 2 correctly. Final Answer: - 699 / 301

Question 14

Evaluate log base 0.125 of 64, i.e., log_{0.125}(64). Write both the power-of-2 reduction and the final numeric value.

Accepted Answer

Introduction / Context: We evaluate a logarithm with base less than 1 by expressing base and argument as powers of a common number (2) and then using exponent division. Given Data / Assumptions: Base: 0.125 = 1/8 = 2^(−3) Argument: 64 = 2^6 Concept / Approach: Use log_{a^m}(a^n) = n/m. For a base less than 1 (negative exponent on 2), the result may be negative even with a > 1 argument. Step-by-Step Solution: log_{0.125}(64) = log_{2^(−3)}(2^6) = 6/(−3) = −2 Verification / Alternative check: Check: (0.125)^(−2) = (1/8)^(−2) = 8^2 = 64, confirming the result. Why Other Options Are Wrong: 2 has the wrong sign for a base less than 1; 0 or “can’t be determined” contradict the exact exponent check above. Common Pitfalls: Ignoring that a fractional base (<1) leads to negative log values for arguments >1 is a common source of sign errors. Final Answer: - 2

Question 15

Interpret and evaluate: log₃₂(8) + log₂₄₃(3^7) − log₃₆(1296). Combine exact values to obtain an integer result.

Accepted Answer

Introduction / Context: Each logarithm can be rewritten using prime powers to obtain exact rational values. The goal is to compute each term exactly and then sum. Given Data / Assumptions: Interpretation: log base 32 of 8, plus log base 243 of 3^7, minus log base 36 of 1296. All logs are well-defined. Concept / Approach: Use a = p^m, b = p^n ⇒ log_a b = n/m. Recognize 32 = 2^5, 8 = 2^3; 243 = 3^5, 3^7 is clear; 1296 = 36^2. Step-by-Step Solution: log₃₂(8) = log_{2^5}(2^3) = 3/5log₂₄₃(3^7) = log_{3^5}(3^7) = 7/5log₃₆(1296) = log_{36}(36^2) = 2Sum: 3/5 + 7/5 − 2 = 10/5 − 2 = 2 − 2 = 0 Verification / Alternative check: Convert each to natural logs: ln 8/ln 32 + ln 3^7/ln 243 − ln 1296/ln 36; simplifying exponents yields the same rational values, confirming the result. Why Other Options Are Wrong: 3, 2, and 1 would require different exponents or bases; the exact arithmetic shows cancellation to zero. Common Pitfalls: Misreading bases or exponents (e.g., treating 243 as 3^4) changes ratios and produces incorrect sums. Carefully factor numbers into prime powers. Final Answer: 0

Question 16

Evaluate the difference: log₄₉(16807) − log₉(27). Convert to prime powers first to obtain an exact rational result.

Accepted Answer

Introduction / Context: We compute two exact logarithms by expressing both bases and arguments as powers of a common prime and then subtract. Given Data / Assumptions: 49 = 7^2, 16807 = 7^5 9 = 3^2, 27 = 3^3 Concept / Approach: Use log_{p^m}(p^n) = n/m. Then subtract the two rational numbers obtained. Step-by-Step Solution: log₄₉(16807) = log_{7^2}(7^5) = 5/2log₉(27) = log_{3^2}(3^3) = 3/2Difference: 5/2 − 3/2 = 1 Verification / Alternative check: Change of base to natural logs gives ln(7^5)/ln(7^2) − ln(3^3)/ln(3^2) = (5/2) − (3/2) as above. Why Other Options Are Wrong: 0 or −1 would require equal or reversed fractions; 3/2 is one of the individual logs, not the difference. Common Pitfalls: Mistaking 16807 as 7^6 or misreading 49/9 bases leads to wrong fractions. Confirm prime-power factorizations first. Final Answer: 1

Question 17

Compute log₉(81) − log₄(32). Express each exactly via prime powers and subtract.

Accepted Answer

Introduction / Context: By expressing each base and argument as a power of a common prime, we can evaluate the logarithms exactly and then subtract. Given Data / Assumptions: 9 = 3^2, 81 = 3^4 4 = 2^2, 32 = 2^5 Concept / Approach: Use log_{p^m}(p^n) = n/m to get exact rational values. Step-by-Step Solution: log₉(81) = log_{3^2}(3^4) = 4/2 = 2log₄(32) = log_{2^2}(2^5) = 5/2 = 2.5Difference: 2 − 5/2 = −1/2 Verification / Alternative check: Change of base to ln confirms the same rational numbers and difference. Why Other Options Are Wrong: 2 is the first term, not the difference; 1/2 has the wrong sign; −3/2 would require 2 − 3.5, which is not the case. Common Pitfalls: Mixing up which number is base and which is argument or reversing the subtraction order changes the sign. Final Answer: - 1 / 2

Question 18

Evaluate the sum of common logs: log 10(10) + log 10(100) + log 10(1000) + log 10(10000) + log 10(100000).

Accepted Answer

Introduction / Context: This is a direct application of the definition of common logarithms on powers of 10, where log 10(10^k) = k. Given Data / Assumptions: Numbers: 10, 100, 1000, 10000, 100000 (i.e., 10^1 to 10^5) Base 10 logarithms. Concept / Approach: Use log 10(10^k) = k and add the results. Step-by-Step Solution: log 10(10) = 1log 10(100) = 2log 10(1000) = 3log 10(10000) = 4log 10(100000) = 5Sum = 1 + 2 + 3 + 4 + 5 = 15 Verification / Alternative check: No alternative needed; the identity is exact for powers of 10. Why Other Options Are Wrong: Expressions like log 11111 or 14·log 100 do not equal the integer sum of the exponents here; only 15 matches. Common Pitfalls: None significant; just ensure all logs are base 10 and recognize each term as a power of 10. Final Answer: 15

Question 19

Given log_10(x) + log_10(y) = z. Express x explicitly in terms of y and z.

Accepted Answer

Introduction / Context: This problem checks your comfort with basic logarithm properties in base 10 (common logs). In particular, it uses the identity log_10(a) + log_10(b) = log_10(a*b). We are asked to isolate x in terms of y and z when the sum of two common logs equals z. Given Data / Assumptions: log_10(x) + log_10(y) = z All quantities are positive reals with x > 0, y > 0 (required for logarithms). Concept / Approach: Combine the sum of logs into a single logarithm, then convert from logarithmic to exponential form to solve for x in terms of y and z. Step-by-Step Solution: log_10(x) + log_10(y) = log_10(xy)So, log_10(xy) = zConvert to exponent form: xy = 10^zHence, x = 10^z / y Verification / Alternative check: Substitute x = 10^z / y back: log_10(10^z / y) + log_10(y) = (log_10(10^z) − log_10(y)) + log_10(y) = z − log_10(y) + log_10(y) = z ✓. Why Other Options Are Wrong: z / y and 10 / z ignore exponentiation implied by the log equation. y / 10^z is the reciprocal of the correct expression. Common Pitfalls: Forgetting that adding logs multiplies arguments, and that solving for a variable inside a log requires converting to exponent form. Final Answer: 10^z / y

Question 20

Evaluate log_{1/3}(81). (Assume real-valued logarithms where the base is positive and not equal to 1.)

Accepted Answer

Introduction / Context: The expression asks for the exponent k such that (1/3)^k = 81. Recognizing 81 as a power of 3 allows a quick evaluation by matching bases and exponents. Given Data / Assumptions: Compute k where (1/3)^k = 81. 81 = 3^4. Concept / Approach: Use base transformation: (1/3)^k = 3^{−k}. Set 3^{−k} equal to 3^4 and equate exponents to solve for k. Step-by-Step Solution: (1/3)^k = 813^{−k} = 3^4−k = 4 ⇒ k = −4 Verification / Alternative check: (1/3)^{−4} = 3^4 = 81 ✓. Why Other Options Are Wrong: 4 changes the sign (would satisfy 3^k = 81, not (1/3)^k = 81). −27 and 127 are unrelated large-magnitude distractors. Common Pitfalls: Forgetting (1/3)^k = 3^{−k}, leading to incorrect sign on the exponent. Final Answer: −4

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