Q: If log(x + 4) = log(4) + log(x) and log(y + 6) = log(6), compare x and y.

Introduction / Context: We solve two simple log equations independently and then compare the resulting values of x and y. The base is common (10 by default) and positivity of arguments must be ensured. Given Data / Assumptions: log(x + 4) = log(4) + log(x) = log(4x). log(y + 6) = log(6). All log arguments are positive. Concept / Approach: Use logarithm properties: log A + log B = log(AB). Since log is one-to-one on positive reals, equate arguments. Step-by-Step Solution: x + 4 = 4x ⇒ 3x = 4 ⇒ x = 4/3 ≈ 1.333…y + 6 = 6 ⇒ y = 0Compare: x ≈ 1.333… > y = 0 ⇒ x > y. Verification / Alternative check: Domain check: x + 4 > 0 and x > 0 (true for x = 4/3). For y, y + 6 > 0 (true for y = 0). Why Other Options Are Wrong: x = y or x y

Question 1

If log(x − 5) = log(x) − log(5) and log(y − 6) = log(y) − log(6), compare x and y.

Accepted Answer

Introduction / Context: This problem tests log laws and equation solving. The identities transform into linear equations in x and y once we use log(a) − log(b) = log(a/b) and apply the one-to-one property of logarithms (same base, positive arguments). Given Data / Assumptions: log(x − 5) = log(x) − log(5) ⇒ log(x − 5) = log(x/5). log(y − 6) = log(y) − log(6) ⇒ log(y − 6) = log(y/6). Log base is 10 (or any common base), arguments positive. Concept / Approach: If log A = log B (same base), then A = B, provided A, B > 0. Solve the resulting linear equations to obtain x and y, then compare. Step-by-Step Solution: x − 5 = x/5 ⇒ 5x − 25 = x ⇒ 4x = 25 ⇒ x = 25/4 = 6.25y − 6 = y/6 ⇒ 6y − 36 = y ⇒ 5y = 36 ⇒ y = 36/5 = 7.2Thus x = 6.25 and y = 7.2, so x < y. Verification / Alternative check: Both solutions satisfy domain constraints: x > 5 and y > 6 so all logs are defined, confirming validity. Why Other Options Are Wrong: “x > y” and “x = y” contradict explicit solutions. “Can’t say” is wrong because the comparison is determined. Common Pitfalls: Forgetting logs require positive arguments and using invalid x ≤ 5 or y ≤ 6. Final Answer: x < y

Question 2

If log(x + 4) = log(4) + log(x) and log(y + 6) = log(6), compare x and y.

Accepted Answer

Introduction / Context: We solve two simple log equations independently and then compare the resulting values of x and y. The base is common (10 by default) and positivity of arguments must be ensured. Given Data / Assumptions: log(x + 4) = log(4) + log(x) = log(4x). log(y + 6) = log(6). All log arguments are positive. Concept / Approach: Use logarithm properties: log A + log B = log(AB). Since log is one-to-one on positive reals, equate arguments. Step-by-Step Solution: x + 4 = 4x ⇒ 3x = 4 ⇒ x = 4/3 ≈ 1.333…y + 6 = 6 ⇒ y = 0Compare: x ≈ 1.333… > y = 0 ⇒ x > y. Verification / Alternative check: Domain check: x + 4 > 0 and x > 0 (true for x = 4/3). For y, y + 6 > 0 (true for y = 0). Why Other Options Are Wrong: x = y or x < y contradict the computed values. “Can’t say” is incorrect; both equations uniquely determine x and y. Common Pitfalls: Using log(a) + log(b) = log(a + b), which is false. Final Answer: x > y

Question 3

If a^x = b, b^y = c, and c^z = a (all positive, a, b, c ≠ 1), find the value of xyz.

Accepted Answer

Introduction / Context: This is a classic log–exponent chain identity. Converting each equation to logarithmic form and multiplying yields a telescoping product that evaluates neatly to 1. Given Data / Assumptions: a, b, c > 0 and not equal to 1 (to avoid degenerate logs). a^x = b, b^y = c, c^z = a. Concept / Approach: Take logarithms to a common base (any base is fine): log_a b, log_b c, log_c a. Use the change-of-base property to relate the factors. Step-by-Step Solution: Let x = log_a b, y = log_b c, z = log_c a.Then xyz = (log_a b)(log_b c)(log_c a).Using change of base: log_b c = (log_a c)/(log_a b), and log_c a = 1/(log_a c).Thus xyz = (log_a b) * [(log_a c)/(log_a b)] * [1/(log_a c)] = 1. Verification / Alternative check: Pick a = 2, b = 4, c = 16: then x = 2, y = 2, z = 1/4 gives xyz = 1, confirming the identity. Why Other Options Are Wrong: 0, 2, 4 contradict the invariant telescoping product. Common Pitfalls: Mistaking log multiplication rules; the telescoping only occurs with change-of-base handled correctly. Final Answer: 1

Question 4

If log_x 4 = 0.4, find the value of x.

Accepted Answer

Introduction / Context: The expression uses a variable base x. The definition log_x 4 = 0.4 means x^0.4 = 4. We solve for x by raising to a reciprocal power or by converting to an exponential equation in base 10 or e. Given Data / Assumptions: log_x 4 = 0.4. x > 0, x ≠ 1 (valid logarithm base). Concept / Approach: By definition, log_x 4 = y ⇔ x^y = 4. Thus x^0.4 = 4 ⇒ x = 4^(1/0.4) = 4^(2.5). Step-by-Step Solution: x = 4^(2.5) = 4^2 * 4^0.5 = 16 * 2 = 32 Verification / Alternative check: Check: log_32 4 = ln 4 / ln 32 = (2 ln 2) / (5 ln 2) = 2/5 = 0.4. Correct. Why Other Options Are Wrong: 4, 16, 1 do not satisfy x^0.4 = 4. Common Pitfalls: Interpreting log x4 as log( x^4 ) in base 10; the intended meaning is base x. Final Answer: 32

Question 5

If log_12 27 = a, find log_6 16 in terms of a.

Accepted Answer

Introduction / Context: We are asked to rewrite one logarithm in terms of another, using change-of-base and prime factorization to express both in terms of ln 2 and ln 3. Setting t = ln 3 / ln 2 simplifies algebraic manipulation. Given Data / Assumptions: a = log_12 27 = ln 27 / ln 12. We want log_6 16 = ln 16 / ln 6. Prime factorizations: 27 = 3^3, 12 = 3 * 2^2, 16 = 2^4, 6 = 2 * 3. Concept / Approach: Let t = ln 3 / ln 2; then a = 3t / (t + 2). Compute log_6 16 = (4 ln 2) / (ln 2 + ln 3) = 4 / (1 + t). Step-by-Step Solution: From a = 3t/(t + 2) ⇒ a(t + 2) = 3t ⇒ 2a = (3 − a) t ⇒ t = 2a / (3 − a)Then log_6 16 = 4 / (1 + t) = 4 / (1 + 2a/(3 − a)) = 4(3 − a)/(3 + a) Verification / Alternative check: Choose numeric a via ln to test; substitution confirms the closed form 4(3 − a)/(3 + a). Why Other Options Are Wrong: Other forms invert signs or reciprocals incorrectly. Common Pitfalls: Mishandling algebra when isolating t from a = 3t/(t + 2). Final Answer: 4(3 - a) / (3 + a)

Question 6

Given log10 57 ≈ 1.756 and log10 0.57 = log10(57) − 2 ≈ −0.244, evaluate: log 57 + log(0.57)^3 + log √(0.57).

Accepted Answer

Introduction / Context: We combine logs using properties: log(a^k) = k log a and log a + log b = log(ab). The expression contains powers of the same positive number (0.57), which we merge systematically. Given Data / Assumptions: log10 57 ≈ 1.756. log10 0.57 = log10 57 − 2 ≈ −0.244. Expression: log 57 + log(0.57)^3 + log √(0.57). Concept / Approach: Interpret log(0.57)^3 as log((0.57)^3) = 3 log(0.57). Similarly, log √(0.57) = (1/2) log(0.57). Step-by-Step Solution: S = log 57 + 3 log 0.57 + (1/2) log 0.57= log 57 + 3.5 log 0.57≈ 1.756 + 3.5(−0.244) = 1.756 − 0.854 ≈ 0.902 Verification / Alternative check: Combine inside a single log: S = log[57 * (0.57)^{3.5}] and evaluate numerically; the same value results to three decimals. Why Other Options Are Wrong: 1.902 and 2.146 add instead of subtracting the negative contributions. 1.146 is from rounding incorrectly. Common Pitfalls: Misreading log(0.57)^3 as (log 0.57)^3; here the intended property is log(a^k). Final Answer: 0.902

Question 7

Evaluate the product (log tan 1°) (log tan 2°) … (log tan 50°).

Accepted Answer

Introduction / Context: Among well-known trigonometric values, tan 45° = 1, so log(tan 45°) = log 1 = 0 (for any base). Because the product explicitly includes the factor with 45°, the entire product collapses to zero. Given Data / Assumptions: Sequence runs from 1° up to 50°; thus it includes 45°. All logs share the same base. Concept / Approach: Multiplying any finite product by 0 yields 0. No other special pairing is required, though identities like tan θ · tan(90° − θ) = 1 can explain complementary behavior. Step-by-Step Reasoning: tan 45° = 1 ⇒ log(tan 45°) = log 1 = 0Product contains a 0 factor ⇒ total product = 0 Verification / Alternative check: Countrate the terms: 1°, 2°, …, 45°, …, 50° includes 45°; hence result is immediate. Why Other Options Are Wrong: 1, 2, −1 ignore the zero factor. Common Pitfalls: Confusing “product of logs” with “log of product”. Here we are literally multiplying log values. Final Answer: 0

Question 8

If the mantissa of log10 3274 is 0.5150, find log10 32.74.

Accepted Answer

Introduction / Context: In common logarithms, log10 N = characteristic + mantissa, where the mantissa depends only on the significant digits (order-of-magnitude independent). Moving the decimal point changes the characteristic but preserves the mantissa. Given Data / Assumptions: log10 3274 has mantissa 0.5150 ⇒ log10 3274 = 3.5150. We need log10 32.74 (same significant digits, decimal shifted left two places). Concept / Approach: Shifting decimal left by k reduces the characteristic by k. Mantissa remains 0.5150 (same significand 3.274 × 10^1 vs 3.274 × 10^3). Step-by-Step Solution: log10 3274 = 3.515032.74 = 3274 × 10^(−2) ⇒ log10 32.74 = 3.5150 − 2 = 1.5150 Verification / Alternative check: Direct estimation: 32.74 lies between 10 and 100, so characteristic must be 1; mantissa stays 0.5150 ⇒ 1.5150. Why Other Options Are Wrong: 2.5150 would correspond to 327.4; 0.5150 would correspond to 3.274. Common Pitfalls: Changing mantissa when only the decimal position changes. Final Answer: 1.5150

Question 9

If log5(x^2 + x) − log5 x = 2, find x.

Accepted Answer

Introduction / Context: Use the quotient law of logarithms to combine the difference of logs, then exponentiate to solve. Ensure arguments are positive and domain constraints are respected (x > 0, x^2 + x > 0). Given Data / Assumptions: log5(x^2 + x) − log5 x = 2. x > 0 and x^2 + x > 0 (automatic for x > 0). Concept / Approach: log5(A) − log5(B) = log5(A/B). If log5 T = 2, then T = 5^2 = 25. Step-by-Step Solution: log5( (x^2 + x)/x ) = 2 ⇒ log5(x + 1) = 2x + 1 = 25 ⇒ x = 24 Verification / Alternative check: Plug in x = 24: LHS = log5(24^2 + 24) − log5(24) = log5(600) − log5(24) = log5(25) = 2. Correct. Why Other Options Are Wrong: 25, 23, 120 do not satisfy x + 1 = 25. Common Pitfalls: Forgetting to simplify (x^2 + x)/x to x + 1. Final Answer: 24

Question 10

If log a, log b, log c are in arithmetic progression (A.P.), what can be said about a, b, c?

Accepted Answer

Introduction / Context: Relating progressions of logarithms to progressions of the original numbers is a standard property: an A.P. in logs corresponds to a G.P. in the numbers because the log function converts products into sums. Given Data / Assumptions: log a, log b, log c are in A.P. ⇒ 2 log b = log a + log c. a, b, c > 0. Concept / Approach: Use log rules: log a + log c = log(ac). Equate and exponentiate if needed. Step-by-Step Reasoning: 2 log b = log a + log c ⇒ log b^2 = log(ac)Therefore, b^2 = ac ⇒ b is the geometric mean of a and cHence a, b, c are in geometric progression (G.P.). Verification / Alternative check: Example: a = 2, c = 8 ⇒ b = √(16) = 4; then log 2, log 4, log 8 differ by equal amounts; numbers 2, 4, 8 are in G.P. Why Other Options Are Wrong: Squares in G.P. is true if and only if a, b, c are in G.P., but the direct, standard statement is that a, b, c themselves are in G.P. A.P. for a, b, c contradicts multiplicative spacing indicated by logs in A.P. Final Answer: a, b, c are in G.P.

Question 11

Find the compound interest on Rs 12,000 for 10 years at 12% per annum, compounded annually.

Accepted Answer

Introduction / Context: Compound interest (CI) accumulates by repeatedly applying the growth factor (1 + r) each compounding period. For annual compounding over n years, Amount A = P(1 + r)^n and CI = A − P. Given Data / Assumptions: Principal P = Rs 12,000. Annual rate r = 12% = 0.12. Years n = 10; annual compounding. Concept / Approach: Compute A = 12000 × (1.12)^10. Then CI = A − 12000. Step-by-Step Solution: (1.12)^5 ≈ 1.76234 ⇒ (1.12)^10 ≈ (1.76234)^2 ≈ 3.105A ≈ 12000 × 3.105 = 37260CI ≈ 37260 − 12000 = 25260 (more precise value ≈ 25270) Verification / Alternative check: Using finer arithmetic (or a calculator) gives (1.12)^10 ≈ 3.10585 ⇒ A ≈ 37270.2 ⇒ CI ≈ 25270.2. The closest listed choice is 25350, matching rounding and typical option design in exams. Why Other Options Are Wrong: 23550 is too low; 2550 ignores compounding depth; 25550 overshoots. Common Pitfalls: Mistaking 10 × 12% simple interest for CI; CI compounds and grows faster than simple interest. Final Answer: 25350

Question 12

How many digits are there in 2^64?

Accepted Answer

Introduction / Context: Digit count uses base-10 logarithms: digits = floor(log10 N) + 1 for positive integers N. We are given N = 2^64, so we only need log10 2 and multiplication. Given Data / Assumptions: N = 2^64. Use log10 2 ≈ 0.30103 (common value). Concept / Approach: Compute log10 N = 64 × log10 2. Apply digits = floor(log10 N) + 1. Step-by-Step Solution: log10(2^64) ≈ 64 × 0.30103 = 19.26592Digits = floor(19.26592) + 1 = 19 + 1 = 20 Verification / Alternative check: 2^10 ≈ 10^3, so 2^60 ≈ 10^18; multiply by 2^4 = 16 gives about 1.6 × 10^19 ⇒ 20 digits. Why Other Options Are Wrong: 18 or 19 underestimate; 21 overestimates the power. Final Answer: 20 digit

Question 13

Given log10 2 = 0.30103 and log10 3 = 0.47712, find the number of digits in 3^12 × 2^8.

Accepted Answer

Introduction / Context: We apply the digit-count formula using given logarithms for 2 and 3. The product's log is the sum of individual logs, scaled by exponents. Given Data / Assumptions: log10 2 = 0.30103, log10 3 = 0.47712. N = 3^12 × 2^8. Concept / Approach: log10 N = 12 log10 3 + 8 log10 2. Digits = floor(log10 N) + 1. Step-by-Step Solution: log10 N = 12(0.47712) + 8(0.30103) = 5.72544 + 2.40824 = 8.13368Digits = floor(8.13368) + 1 = 8 + 1 = 9 Verification / Alternative check: Since 10^8 < N < 10^9, there must be 9 digits. Why Other Options Are Wrong: 6, 7, 8 underestimate the magnitude. Final Answer: 9

Question 14

Evaluate 1/(log_2 π) + 1/(log_6 π).

Accepted Answer

Introduction / Context: Use the reciprocal log identity: 1/log_a b = log_b a (a, b > 0, a, b ≠ 1). This lets us express the sum in a common base and combine using log addition rules. Given Data / Assumptions: Terms: 1/(log_2 π) and 1/(log_6 π). π ≈ 3.1416; bases 2 and 6 are valid (≠ 1). Concept / Approach: 1/log_a b = log_b a ⇒ 1/log_2 π = log_π 2; 1/log_6 π = log_π 6. Sum ⇒ log_π 2 + log_π 6 = log_π (12). Step-by-Step Solution: S = log_π 2 + log_π 6 = log_π (12)Since 12 > π (≈ 3.1416), log_π (12) > 1 Verification / Alternative check: Compute approximately: ln 12 / ln π ≈ 2.4849 / 1.1447 ≈ 2.17 > 1. Why Other Options Are Wrong: “Less than 1” contradicts 12 > π. “Between 5 and 6” grossly overestimates. “None of these” is incorrect since “greater than 1” is true. Final Answer: greater then 1

Question 15

Find the number of digits in (875)^16.

Accepted Answer

Introduction / Context: We apply digit counting via base-10 logarithms. For N > 0, digits = floor(log10 N) + 1. Use factorization to compute log10 875 accurately. Given Data / Assumptions: N = (875)^16. 875 = 7 × 125 = 7 × 5^3. Concept / Approach: log10 875 = log10(7) + 3 log10(5). Then log10 N = 16 · log10 875; digits follow from flooring + 1. Step-by-Step Solution: log10 7 ≈ 0.845098, log10 5 ≈ 0.69897log10 875 ≈ 0.845098 + 3(0.69897) = 0.845098 + 2.09691 = 2.942008log10 N ≈ 16 × 2.942008 = 47.072128Digits = floor(47.072128) + 1 = 47 + 1 = 48 Verification / Alternative check: (1000)^16 has 49 digits; since 875 < 1000, (875)^16 has fewer ⇒ 48 digits fits the estimate. Why Other Options Are Wrong: 47 undercounts (would require N < 10^46). 49 or 50 overcount relative to (1000)^16. Final Answer: 48 digit

Question 16

A town has population 2,65,000 at the beginning of 1986. If the annual growth rate is 52 per thousand (i.e., 5.2% per year), find the population at the beginning of 1991.

Accepted Answer

Introduction / Context: Year-on-year percentage growth compounds. Over 5 years (from the start of 1986 to the start of 1991), the population multiplies by (1 + r)^5 with r = 5.2% = 0.052. Given Data / Assumptions: Initial population P0 = 2,65,000. Annual growth rate r = 0.052 (52 per thousand). Years n = 5. Concept / Approach: Population after n years: Pn = P0(1 + r)^n. Round to the nearest hundred as per common exam convention if needed. Step-by-Step Solution: (1.052)^2 ≈ 1.1067; (1.052)^4 ≈ 1.1067^2 ≈ 1.2248; multiply by 1.052 ⇒ (1.052)^5 ≈ 1.289P5 ≈ 2,65,000 × 1.289 ≈ 3,41,585Closest listed population ≈ 3,41,400 Verification / Alternative check: Using a more precise value (1.052)^5 ≈ 1.28936 gives ≈ 3,41,690; among the options, 3,41,400 is the nearest standard rounded choice. Why Other Options Are Wrong: 3,40,400 and 3,42,400 are farther from the precise computation; 3,43,400 overshoots. Common Pitfalls: Treating the growth as linear (simple increase) instead of compound. Final Answer: 3,41,400

Question 17

Given that log 2 = 0.3010 and log 3 = 0.4771 (logarithms to base 10), find the value of log base 5 of 512, that is, log₅ 512.

Accepted Answer

Introduction / Context: This question tests your understanding of logarithms, especially change of base and how to work with given common logarithm values. You are asked to compute log base 5 of 512 using only the given values log 2 and log 3 to base 10. This mimics typical exam conditions where you must manipulate logs algebraically rather than relying on a calculator. Given Data / Assumptions: log 2 = 0.3010 (base 10). log 3 = 0.4771 (base 10). We need to find log₅ 512. Logarithms are common logs (base 10) when no base is specified. Concept / Approach: We use three main ideas. First, express 512 as a power of 2. Second, write log base 5 as a ratio of common logarithms using the change of base formula: log₅ N = (log N) / (log 5). Third, express log 5 in terms of log 2, because 5 = 10 / 2. This allows us to compute log 5 using 1 and log 2, which are known. Finally, substitute and simplify numerically to select the closest matching option. Step-by-Step Solution: Observe that 512 = 2⁹. Using the change of base formula, log₅ 512 = log 512 / log 5. Compute log 512 using powers of 2: log 512 = log(2⁹) = 9 log 2 = 9 × 0.3010 = 2.709. Next, find log 5. Since 10 = 2 × 5, we have log 10 = log 2 + log 5. log 10 = 1, so 1 = 0.3010 + log 5, giving log 5 = 1 - 0.3010 = 0.6990. Now compute log₅ 512 = 2.709 / 0.6990. The ratio 2.709 / 0.6990 is approximately 3.875 (close to 3.874, rounded). Verification / Alternative check: We know 5³ = 125 and 5⁴ = 625, so 512 lies between 5³ and 5⁴. Therefore log₅ 512 must lie between 3 and 4. Among the options, 3.875 is the only value between 3 and 4 and close to our calculated approximation, confirming our answer. Why Other Options Are Wrong: 2.875: Less than 3, but 512 is larger than 125 = 5³, so the logarithm must be greater than 3. 4.875 and 5.875: These are larger than 4, which would correspond to numbers greater than 625 = 5⁴, so they cannot represent log₅ 512. Common Pitfalls: Students sometimes forget the change of base formula or mistakenly divide in the wrong order (log 5 / log 512). Another frequent error is computing log 5 incorrectly, for example using log 5 = log 2 + 1 instead of 1 - log 2. Rounding too early can also give slightly off answers, but the approximate range check between 3 and 4 usually corrects such issues. Final Answer: The required value is 3.875.

Question 18

If log 2 (to base 10) is 0.30103, how many digits are there in the number 2^56?

Accepted Answer

Introduction / Context: This question checks your understanding of how to use logarithms to determine the number of digits in a large power of a number. Instead of directly computing 2^56, which is unwieldy, we use the relationship between common logarithms and decimal digit counts. Given Data / Assumptions: log 2 = 0.30103 (base 10). We need the number of digits in 2^56. All logs are common logarithms (base 10). Concept / Approach: If a positive integer N has common logarithm log N = L, then the number of digits of N in base 10 is given by floor(L) + 1. We therefore compute log(2^56) using laws of logarithms, then apply this digit formula. The key rules are log(a^n) = n log a and the floor plus one rule for digit count. Step-by-Step Solution: We want the number of digits in N = 2^56. Compute log N = log(2^56) = 56 log 2. Given log 2 = 0.30103, we have log N = 56 × 0.30103. Multiply: 0.30103 × 50 = 15.0515 and 0.30103 × 6 = 1.80618. Add them: 15.0515 + 1.80618 ≈ 16.85768. So log(2^56) ≈ 16.85768. The number of digits in N is floor(16.85768) + 1 = 16 + 1 = 17. Verification / Alternative check: We know 2^10 ≈ 10^3 (since 2^10 = 1024). Then 2^50 ≈ 10^15 and 2^6 = 64. So 2^56 = 2^50 × 2^6 ≈ 10^15 × 64 ≈ 6.4 × 10^16, which clearly has 17 digits. This quick estimation matches the detailed logarithmic computation. Why Other Options Are Wrong: 19, 23, 25: These values would correspond to much larger exponents. For example, a 19-digit number would be around 10^18, while our estimate shows 2^56 is of the order 10^16 to 10^17. Therefore all these options are too large for the digit count. Common Pitfalls: Some learners mistakenly take the integer part of the logarithm without adding one, which would give 16 instead of 17. Others miscalculate 56 × 0.30103 or round too roughly. It is important to remember the precise rule: number of digits = floor(log₁₀ N) + 1 for positive integers. Final Answer: The number 2^56 has 17 digits.

Question 19

If log 27 (to base 10) is 1.431, then what is the value of log 9 (to base 10)?

Accepted Answer

Introduction / Context: This problem checks your familiarity with the laws of logarithms and the ability to express logs of related numbers in terms of each other. You are given log 27 and asked to find log 9, both with the same base, by recognising that 27 and 9 are powers of 3. Given Data / Assumptions: log 27 = 1.431 (base 10). We need log 9 to the same base. 27 = 3³ and 9 = 3². Concept / Approach: The central idea is to rewrite each number as a power of the same base, here base 3, and then use the rule log(a^k) = k log a. Since 27 and 9 share the same prime base, 3, we can solve for log 3 first using the given information and then multiply it by 2 to find log 9. Step-by-Step Solution: Write 27 as 3³. Then log 27 = log(3³) = 3 log 3. We are told log 27 = 1.431, so 3 log 3 = 1.431. Hence log 3 = 1.431 / 3. Compute the division: 1.431 / 3 = 0.477. Now express 9 as 3². Then log 9 = log(3²) = 2 log 3. Substitute log 3: log 9 = 2 × 0.477 = 0.954. Verification / Alternative check: We know approximately that log 3 is around 0.4771 in standard tables. Doubling that gives about 0.9542, which agrees closely with the calculated 0.954 and matches the given multiple-choice option. So the answer is consistent with known values. Why Other Options Are Wrong: 0.754 and 0.654: These values are too small and would correspond to squares of numbers less than 3. 0.854: This is also smaller than the correct value. Since 9 is greater than 7 and less than 10, its logarithm should be between log 7 and 1, which suggests a value closer to 0.95 than to 0.85. Common Pitfalls: A common mistake is to divide 1.431 by 2 instead of 3 when solving for log 3, or to forget that log 9 equals 2 log 3, not log 3² treated in some other way. Another error is attempting to approximate logs of 9 directly without using the relationship with log 3, which is slower and less accurate. Final Answer: The value of log 9 is 0.954.

Question 20

If logₐ (a b) = x, where a and b are positive and a ≠ 1, then what is the value of log_b (a b) in terms of x?

Accepted Answer

Introduction / Context: This is a conceptual logarithm question that asks you to relate two different logarithmic expressions involving the same product a b but different bases. It tests your understanding of the change of base idea and the relationship between a logarithm and its reciprocal when bases are swapped. Given Data / Assumptions: logₐ (a b) = x. a and b are positive real numbers with a ≠ 1 and b ≠ 1. We must find log_b (a b) expressed in terms of x. Concept / Approach: From the equation logₐ (a b) = x we can convert to exponential form and express b in terms of a and x. Then we find log_b (a b) using the definition of logarithms and the property that log_b a is the reciprocal of logₐ b. Combining these relationships allows us to express the required logarithm purely in terms of x. Step-by-Step Solution: Start with logₐ (a b) = x. By definition, this means a^x = a b. Divide both sides by a: b = a^(x - 1). We need log_b (a b). By logarithm rules, log_b (a b) = log_b a + log_b b. Note that log_b b = 1. Next, log_b a = 1 / logₐ b (since logs with swapped bases are reciprocals). From b = a^(x - 1), we have logₐ b = x - 1. Thus log_b a = 1 / (x - 1). Therefore, log_b (a b) = log_b a + 1 = 1 / (x - 1) + 1. Combine the terms: 1 / (x - 1) + 1 = (1 + x - 1) / (x - 1) = x / (x - 1). Verification / Alternative check: We can test with a simple numerical example. Let a = 2 and let b = 4. Then a b = 8. Compute log₂ 8 = 3, so x = 3. According to our formula, log_b (a b) = x / (x - 1) = 3 / 2. Indeed, log₄ 8 = log 8 / log 4 = 3 log 2 / (2 log 2) = 3 / 2, which matches perfectly. Why Other Options Are Wrong: 1/x: This ignores the effect of the added factor a inside the logarithm and does not match the test example. x/(x+1) and x/(1-x): Both give incorrect values when tested with simple numerical choices of a and b, and they do not follow from the algebraic steps above. Common Pitfalls: Learners often forget to convert the initial logarithmic equation into exponential form or mishandle the reciprocal relationship between logₐ b and log_b a. Missing the step log_b (a b) = log_b a + 1 is another common error, as is algebraic slippage when simplifying 1 / (x - 1) + 1. Final Answer: The required value is x / (x - 1).

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