Question 1

If log_10{ log_10[ log_10( log_10 N ) ] } = 0, find N.

Accepted Answer

Introduction / Context: Nesting logarithms requires peeling them one by one from the outside in. The key fact is log_10(A) = 0 implies A = 1. Applying this repeatedly determines the inner values stepwise until N is obtained. Given Data / Assumptions: log_10{ log_10[ log_10( log_10 N ) ] } = 0. All log arguments must be positive. Concept / Approach: Let the innermost value be t_1 = log_10 N. Then set t_2 = log_10(t_1), t_3 = log_10(t_2). The given states log_10(t_3) = 0 ⇒ t_3 = 1. Work backward to find N. Step-by-Step Solution: log_10(t_3) = 0 ⇒ t_3 = 1But t_3 = log_10(t_2) ⇒ log_10(t_2) = 1 ⇒ t_2 = 10t_2 = log_10(t_1) ⇒ log_10(t_1) = 10 ⇒ t_1 = 10^{10}t_1 = log_10 N ⇒ log_10 N = 10^{10} ⇒ N = 10^{(10^{10})} Verification / Alternative check: Substitute forward: log_10 N = 10^{10}; log_10(10^{10}) = 10; log_10(10) = 1; log_10(1) = 0 ✓. Why Other Options Are Wrong: 10^10 and 10^100 are intermediate inner values (t_1 or a confusion), not N; 10^1010 misreads exponent placement. Common Pitfalls: Losing track of nesting levels or assuming log_10(0)=… (undefined). Maintain positivity of each inner argument. Final Answer: 10^(10^10)

Question 2

Evaluate 25^{log_5 4}.

Accepted Answer

Introduction / Context: Powers of logs often simplify via base-matching identities. Here 25 is a power of 5, and the exponent involves log base 5, which suggests rewriting to use a^log_a(b) = b. Given Data / Assumptions: Compute 25^{log_5 4}, noting 25 = 5^2. Concept / Approach: Use exponent rules: (5^2)^{log_5 4} = 5^{2 * log_5 4}. Then apply the identity 5^{log_5 4} = 4 to simplify. Step-by-Step Solution: 25^{log_5 4} = (5^2)^{log_5 4}= 5^{2 * log_5 4}= (5^{log_5 4})^2 = 4^2 = 16 Verification / Alternative check: Numerically, log_5 4 ≈ 0.861... Then 25^{0.861...} ≈ 16 ✓. Why Other Options Are Wrong: 5 or 25 result from misapplying a^log_a(b) = b or forgetting the outer square. Common Pitfalls: Applying log identity directly to 25 rather than rewriting it as 5^2 first. Final Answer: 16

Question 3

Find log_10(0.000001).

Accepted Answer

Introduction / Context: Negative powers of 10 translate directly to negative common logs, because log_10(10^k) = k for any real k. Given Data / Assumptions: 0.000001 = 10^{−6}. Concept / Approach: Apply the fundamental log identity log_10(10^k) = k with k = −6. Step-by-Step Solution: log_10(0.000001) = log_10(10^{−6}) = −6 Verification / Alternative check: Move decimal six places to the right multiplies by 10^6; the log therefore drops by 6. Why Other Options Are Wrong: 6 and ±5 correspond to incorrect exponents. Common Pitfalls: Confusing the number of zeros; note there are six places after the decimal before the 1. Final Answer: −6

Question 4

Find log_10(0.00001).

Accepted Answer

Introduction / Context: Recognize small decimal powers as powers of 10 to evaluate common logarithms directly. Given Data / Assumptions: 0.00001 = 10^{−5}. Concept / Approach: Use log_10(10^k) = k. Step-by-Step Solution: log_10(0.00001) = log_10(10^{−5}) = −5 Why Other Options Are Wrong: −6, −7 represent incorrect counting of decimal places. Final Answer: −5

Question 5

Evaluate log_6( log_5(15625) ).

Accepted Answer

Introduction / Context: This tests composition of logs with different bases. Simplify the inner logarithm first by recognizing perfect powers, then apply the outer logarithm. Given Data / Assumptions: 15625 is a power of 5. Concept / Approach: Compute t = log_5(15625). Since 15625 = 5^6, t = 6. Then evaluate log_6(6) = 1. Step-by-Step Solution: 15625 = 5^6 ⇒ log_5(15625) = 6log_6(6) = 1 Why Other Options Are Wrong: 2 or 3 result from misidentifying the power of 5 or misapplying the outer log. Final Answer: 1

Question 6

Given log_{10,000}(x) = −1/4, find x.  (Repaired example for clarity; see similar style above.)

Accepted Answer

Note: This placeholder entry documents the repaired-notation approach applied to ambiguous stems elsewhere in the set. See the solved item with id 100808 for the final, fully populated update.

Question 7

If log_x(16/25) = −1/2, find x.

Accepted Answer

Introduction / Context: When the base is unknown, convert the logarithmic equation to exponential form. A negative log means the base raised to a negative exponent equals the argument; equivalently, the reciprocal of a square root relation often appears with −1/2. Given Data / Assumptions: log_x(16/25) = −1/2. Concept / Approach: Use log definition: log_x(A) = y ⇔ x^y = A. Here, x^{−1/2} = 16/25 ⇒ (1/√x) = 16/25 ⇒ √x = 25/16 ⇒ x = (25/16)^2. Step-by-Step Solution: x^{−1/2} = 16/251/√x = 16/25 ⇒ √x = 25/16x = (25/16)^2 = 625/256 Verification / Alternative check: Compute log_{625/256}(16/25) = −1/2 using change of base to confirm numerically. Why Other Options Are Wrong: 256/625 is the reciprocal; 526/265 is a distractor with swapped digits. Final Answer: 625 / 256

Question 8

Find log_2(1/64).

Accepted Answer

Introduction / Context: 1/64 is a reciprocal power of 2. Logs with base 2 translate powers of 2 into their exponents directly. Given Data / Assumptions: 1/64 = 2^{−6}. Concept / Approach: Use log_2(2^k) = k. Step-by-Step Solution: log_2(1/64) = log_2(2^{−6}) = −6 Final Answer: −6

Question 9

If log_a(3) = 1/3, find a.

Accepted Answer

Introduction / Context: Convert from logarithmic to exponential form to find the base a that makes the statement true. Given Data / Assumptions: log_a(3) = 1/3. Concept / Approach: By definition, a^{1/3} = 3 ⇒ a = 3^3. Step-by-Step Solution: a^{1/3} = 3 ⇒ a = 27 Final Answer: 27

Question 10

If log_10(x) = 7, compute x.

Accepted Answer

Introduction / Context: A common logarithm equal to 7 means the number is 10 raised to the 7th power (antilogarithm). Given Data / Assumptions: log_10(x) = 7. Concept / Approach: Use the definition: log_10(x) = 7 ⇔ x = 10^7. Final Answer: 10^7

Question 11

If p^x = q, which logarithmic relation is correct?

Accepted Answer

Introduction / Context: Exponentials and logarithms are inverses. Translating p^x = q into a logarithmic statement tests understanding of this inverse relationship. Given Data / Assumptions: p > 0, p ≠ 1, q > 0; p^x = q. Concept / Approach: By definition, log base p of q equals the exponent x that raises p to q. Step-by-Step Solution: p^x = q ⇔ x = log_p(q) Why Other Options Are Wrong: They mismatch positions of base/argument or set the log equal to the base rather than the exponent. Final Answer: log_p q = x

Question 12

Given log_10(2) = 0.3010, find log_10(5).

Accepted Answer

Introduction / Context: Use the identity 10 = 2 * 5 and the logarithm of a product to relate log(2) and log(5). Given Data / Assumptions: log_10(2) = 0.3010 and log_10(10) = 1. Concept / Approach: Since log(10) = log(2) + log(5), we have log(5) = 1 − log(2). Step-by-Step Solution: log_10(5) = 1 − 0.3010 = 0.6990 Final Answer: 0.6990

Question 13

If log_10(10x) = 2.7532, find log_10(10000x).

Accepted Answer

Introduction / Context: Multiplying a positive number by a power of 10 increases its common logarithm by the exponent (the characteristic). This is standard in mantissa–characteristic operations. Given Data / Assumptions: log_10(10x) = 2.7532 ⇒ log_10(x) = 1.7532. Concept / Approach: Compute log_10(10000x) = log_10(x) + 4 because 10000 = 10^4. Step-by-Step Solution: log_10(x) = 2.7532 − 1 = 1.7532log_10(10000x) = 1.7532 + 4 = 5.7532 Final Answer: 5.7532

Question 14

If (log 125) / (log 5) = x (same log base in numerator and denominator), find x.

Accepted Answer

Introduction / Context: A ratio of logarithms with the same base equals a logarithm with the base changed (change-of-base formula). This simplifies to a log with base 5 of 125. Given Data / Assumptions: x = (log 125)/(log 5), any consistent base for log. Concept / Approach: x = log_5(125) because (log_b A)/(log_b C) = log_C A. Step-by-Step Solution: 125 = 5^3 ⇒ log_5(125) = 3 Final Answer: 3

Question 15

Compute (log_5 3) × (log_3 625).

Accepted Answer

Introduction / Context: The product log_a b · log_b c equals log_a c by cancellation of the intermediate base. Apply that here with a = 5, b = 3, c = 625. Given Data / Assumptions: 625 = 5^4. Concept / Approach: (log_5 3)(log_3 625) = log_5 625 = 4. Step-by-Step Solution: (log_5 3)(log_3 625) = (ln 3/ln 5)(ln 625/ln 3) = ln 625/ln 5 = log_5 625Since 625 = 5^4 ⇒ log_5 625 = 4 Final Answer: 4

Question 16

Evaluate log_9(27) − log_{27}(9).

Accepted Answer

Introduction / Context: Translate both logs to a common base (e.g., natural log) or use base–exponent recognition to evaluate each term exactly, then subtract. Given Data / Assumptions: 27 = 3^3 and 9 = 3^2. Concept / Approach: Compute each term via ln: log_9(27) = ln 27 / ln 9 = (3 ln 3)/(2 ln 3) = 3/2. Similarly, log_{27}(9) = ln 9 / ln 27 = (2 ln 3)/(3 ln 3) = 2/3. Step-by-Step Solution: log_9(27) − log_{27}(9) = 3/2 − 2/3 = (9/6 − 4/6) = 5/6 Final Answer: 5 / 6

Question 17

If 10^x = 1.73 and log_10(1730) = 3.2380 (so antilog(0.2380) = 1.73), find x.

Accepted Answer

Introduction / Context: This tests understanding of characteristics and mantissas in common logarithms. If log_10(1730) = 3.2380, the mantissa 0.2380 corresponds to the significant 1.73 because 1730 = 1.73 × 10^3. Given Data / Assumptions: 10^x = 1.73 and log_10(1730) = 3.2380 ⇒ antilog(0.2380) = 1.73. Concept / Approach: From 10^x = 1.73 it follows directly that x = log_10(1.73). The provided log indicates log_10(1.73) = 0.2380 (mantissa retained). Step-by-Step Solution: x = log_10(1.73) = 0.2380 Why Other Options Are Wrong: 1.2380 and 2.2380 include an added characteristic (×10 or ×100); 2.380 drops a zero and changes scale. Final Answer: 0.2380

Question 18

Evaluate (log(a^n)) / (log(a^{bn})) and express the result in simplest symbolic form (a > 0, a ≠ 1; b, n ≠ 0).

Accepted Answer

Introduction / Context: Simplify a ratio of logarithms with the same base and power arguments. Using log rules, powers can be pulled down as multipliers. Many expressions collapse cleanly when the same base and variable powers are present in numerator and denominator. Given Data / Assumptions: a > 0, a ≠ 1 (log base conditions). b ≠ 0, n ≠ 0. Expression: (log(a^n)) / (log(a^{bn})). Concept / Approach: Use log power rule: log(a^k) = k * log(a) for any admissible base (here the unspecified common base cancels). Then simplify the scalar ratio. Step-by-Step Solution: log(a^n) = n * log(a)log(a^{bn}) = (bn) * log(a)(log(a^n)) / (log(a^{bn})) = [n * log(a)] / [bn * log(a)] = 1 / b Verification / Alternative check: Pick concrete values, e.g., a = 10, n = 2, b = 3. Then (log(10^2))/(log(10^6)) = 2/6 = 1/3 = 1/b ✓. Why Other Options Are Wrong: They introduce extraneous terms or swap bases; the ratio purely cancels to 1/b by the power rule and common factor log(a). Common Pitfalls: Confusing log base with argument; forgetting both numerator and denominator share the same multiplicative factor log(a). Final Answer: 1 / b

Question 19

Find the number of digits in 8^10. (Given log10 2 = 0.3010)

Accepted Answer

Introduction / Context: Digit count problems rely on logarithms. If N > 0, then the number of decimal digits is floor(log10 N) + 1. We are given log10 2, which lets us evaluate powers of 2 efficiently. Given Data / Assumptions: N = 8^10. 8 = 2^3, so 8^10 = (2^3)^10 = 2^30. log10 2 = 0.3010 (approximation sufficient for digit counting). Concept / Approach: Compute log10(2^30) = 30 * log10 2. Use the digit-count formula: digits = floor(log10 N) + 1. Step-by-Step Solution: log10(8^10) = log10(2^30) = 30 * 0.3010 = 9.03Digits = floor(9.03) + 1 = 9 + 1 = 10 Verification / Alternative check: Since 10^9 = 1,000,000,000 and 10^10 = 10,000,000,000, and 8^10 ≈ 1.07 × 10^9, it lies between 10^9 and 10^10, confirming 10 digits. Why Other Options Are Wrong: 19, 17, 20 misapply the digit formula or misuse log10 2. “None of these” is incorrect because 10 is exact. Common Pitfalls: Forgetting to add 1 after flooring log10 N. Final Answer: 10

Question 20

Find the number of digits in 8^57. (Given log10 2 = 0.3010)

Accepted Answer

Introduction / Context: We again use the digit-count identity: digits = floor(log10 N) + 1. Express 8^57 as a power of 2 to leverage the provided log10 2 value. Given Data / Assumptions: N = 8^57 = (2^3)^57 = 2^171. log10 2 = 0.3010. Concept / Approach: Compute log10(2^171) = 171 * 0.3010. Apply digits = floor(log10 N) + 1. Step-by-Step Solution: log10 N = 171 * 0.3010 = 51.471Digits = floor(51.471) + 1 = 51 + 1 = 52 Verification / Alternative check: Since 10^51 < N < 10^52 and it is not an exact power of 10, the digit count must be 52. Why Other Options Are Wrong: 50, 51, 53 result from rounding errors or dropping the +1. Common Pitfalls: Using 0.3 instead of 0.3010 gives 51.3 and still yields 52 digits, but coarser approximations can mislead if rounding is mishandled. Final Answer: 52

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