Question 1

Find the logarithm of 144 to base 23, that is, evaluate log₂₃ 144 and choose the closest correct option.

Accepted Answer

Introduction / Context: This problem is about estimating the value of a logarithm with an unusual base. Instead of asking for a detailed decimal approximation, the question simply wants to know which listed option could be equal to log₂₃ 144. You can solve this by comparing powers of 23 with 144 without using a calculator. Given Data / Assumptions: We need log₂₃ 144. Base is 23, argument is 144. Available options are 2, 4, 8, and None of these. All numbers are positive, and the logarithm is real. Concept / Approach: The logarithm log₂₃ 144 answers the question, "To what power must 23 be raised to obtain 144?" Therefore, we should compare 23¹, 23², and nearby powers to see the range in which 144 falls. If 144 were exactly a simple power of 23, its logarithm would be an integer, but otherwise it will be some non-integer value between two integers. This reasoning alone is enough to choose the correct option from the list. Step-by-Step Solution: Start with 23¹ = 23. Next, compute 23² = 23 × 23 = 529. Now compare 144 with these powers. We see that 23 < 144 < 529. Therefore, 23¹ < 144 < 23². This implies 1 < log₂₃ 144 < 2, because logarithms are increasing functions for bases greater than 1. Hence log₂₃ 144 is not equal to 2, 4, or 8, all of which are integers outside the interval (1, 2). Verification / Alternative check: If log₂₃ 144 were exactly 2, then 23² would equal 144, but 23² is 529. If it were 4, then 23⁴ would need to equal 144, but 23⁴ is huge, far above 144. Thus, any integer value from the options is clearly impossible. Why Other Options Are Wrong: 2: Would require 144 = 23² = 529, which is false. 4: Would require 144 = 23⁴, which is vastly larger than 144. 8: Would require 144 = 23⁸, an even more enormous number, so this is clearly incorrect. Common Pitfalls: Sometimes students try to overcomplicate such questions by attempting long logarithm calculations without real need. Here, conceptual understanding of logarithm behaviour and simple power comparisons are enough. Another mistake is to assume that because the number 144 is a perfect square (12²), it must somehow match a simple integer logarithm base 23, which is not true. Final Answer: Therefore, none of the integer values listed match log₂₃ 144. The correct choice is None of these.

Question 2

If log 64 (to base 10) is 1.8061, then what is the approximate value of log 16 (to base 10)?

Accepted Answer

Introduction / Context: This is a straightforward logarithm problem that uses the laws of indices and logarithms. You are given the common logarithm of 64 and asked to deduce the logarithm of 16. Since both numbers are powers of 2, the connection between them can be exploited to find log 16 without using tables or a calculator directly. Given Data / Assumptions: log 64 = 1.8061 (base 10). We need log 16 to the same base. 64 = 2⁶ and 16 = 2⁴. Concept / Approach: We make use of the identity log(a^k) = k log a. Since 64 and 16 are powers of the same base, we can express both in terms of log 2 and relate log 16 to log 64. Express log 64 as 6 log 2, then find log 2 and use it to compute log 16 as 4 log 2. Step-by-Step Solution: Write 64 as 2⁶, so log 64 = log(2⁶) = 6 log 2. We are given 6 log 2 = 1.8061. Therefore, log 2 = 1.8061 / 6. Divide: 1.8061 / 6 ≈ 0.3010167. Next, write 16 as 2⁴. So log 16 = log(2⁴) = 4 log 2. Substitute log 2: log 16 = 4 × 0.3010167 ≈ 1.2040668. Rounded to four decimal places, log 16 ≈ 1.2040. Verification / Alternative check: We know that 10¹ = 10 and 10² = 100. Since 16 lies between 10 and 100, log 16 should lie between 1 and 2. A value of about 1.2 is therefore reasonable. This cross-check with the position of 16 on the number line confirms that 1.2040 is a sensible approximation. Why Other Options Are Wrong: 1.9048 and 1.4521: These are too large for log 16, because they correspond to numbers much closer to 80 or 90 or beyond, not 16. 0.9840: This is less than 1, which would correspond to a number less than 10, whereas 16 is greater than 10. Common Pitfalls: A common mistake is to take a simple ratio of 16 and 64 and attempt to apply it directly to the logarithms without using the power rule. Another pitfall is rounding too early; dividing 1.8061 coarsely may introduce more error than necessary, although in this problem the answer options are far enough apart that small rounding differences do not affect the choice. Final Answer: The approximate value of log 16 is 1.2040.

Question 3

A fast moving superfast express train crosses a slower passenger train in 20 seconds. The speed of the faster train is 72 km/h and the speed of the slower train is 27 km/h. The length of the faster train is 100 metres. If both trains are moving in t…

Accepted Answer

Introduction / Context: This question is a standard relative speed and trains problem. When one train overtakes another, the time taken depends on their relative speed and the total distance that must be covered, which is the sum of their lengths. We are given the speeds, the time of overtaking, and the length of the faster train, and must determine the length of the slower train. Given Data / Assumptions: Speed of the faster train = 72 km/h. Speed of the slower train = 27 km/h. Length of the faster train = 100 metres. Time taken to completely cross the slower train = 20 seconds. Both trains move in the same direction on parallel tracks. Concept / Approach: When two bodies move in the same direction, their relative speed is the difference of their speeds. During overtaking, the faster train has to cover a distance equal to the sum of the lengths of both trains with respect to the slower train. We convert the speeds from km/h to m/s, compute the relative speed, multiply by time to get the total distance covered, and then subtract the known length of the faster train to obtain the length of the slower train. Step-by-Step Solution: Convert speeds to m/s using 1 km/h = 5/18 m/s. Speed of faster train = 72 × 5/18 = 20 m/s. Speed of slower train = 27 × 5/18 = 7.5 m/s. Relative speed when moving in the same direction = 20 - 7.5 = 12.5 m/s. Time taken to overtake = 20 seconds. Total distance covered relative to the slower train = relative speed × time = 12.5 × 20 = 250 metres. This total distance equals the sum of the lengths of both trains. Let L be the length of the slower train. Then 100 + L = 250. So L = 250 - 100 = 150 metres. Verification / Alternative check: We can verify by reversing the reasoning. If the slower train is 150 metres long, then combined length = 100 + 150 = 250 metres. At a relative speed of 12.5 m/s, time to cross is distance / speed = 250 / 12.5 = 20 seconds, matching the given value. So the answer is consistent. Why Other Options Are Wrong: 100 m: Would imply combined length 200 metres, giving time = 200 / 12.5 = 16 seconds, not 20 seconds. 125 m and 175 m: These lengths would produce combined distances of 225 or 275 metres, which lead to overtaking times of 18 seconds or 22 seconds, neither matching the given 20 seconds. Common Pitfalls: Common mistakes include adding speeds instead of subtracting them for same-direction motion, forgetting to convert km/h to m/s, or mistakenly treating the distance to be covered as only one train length instead of the sum of both lengths. Always identify whether trains move in the same or opposite directions and convert units before applying formulas. Final Answer: The length of the slower passenger train is 150 metres.

Question 4

For a natural number x > 1, let p = logₓ (x + 1) and q = log₍ₓ₊₁₎ (x + 2). Which one of the following is correct about p and q?

Accepted Answer

Introduction / Context: This problem is a conceptual logarithm inequality question. Instead of asking for a numeric value, it compares two expressions involving logarithms with slightly different bases and arguments. Such questions test your understanding of how logarithmic functions behave as the base and argument change, and they are common in higher level aptitude and entrance exams. Given Data / Assumptions: x is a natural number with x > 1. p = logₓ (x + 1). q = log₍ₓ₊₁₎ (x + 2). We must decide whether p < q, p = q, p > q, or if the relation cannot be determined. Concept / Approach: For bases greater than 1, logarithmic functions are increasing in their argument and decreasing in their base. In p, the base is x and the argument is x + 1. In q, the base has increased to x + 1 and the argument to x + 2. We can compare p and q by writing both in terms of natural logarithms and carefully analysing how the ratios behave. Another powerful tool is to test several values of x to observe a pattern and then reason more generally. Step-by-Step Solution: Write p and q using natural logs: p = ln(x + 1) / ln x and q = ln(x + 2) / ln(x + 1). Evaluate numerically for x = 2. Then p = ln 3 / ln 2 ≈ 1.585 and q = ln 4 / ln 3 ≈ 1.262. So p > q for x = 2. For x = 3, p = ln 4 / ln 3 ≈ 1.262 while q = ln 5 / ln 4 ≈ 1.113, again p > q. For x = 4, p = ln 5 / ln 4 ≈ 1.161 while q = ln 6 / ln 5 ≈ 1.113, still p > q. These consistent results suggest that p is greater than q for all natural x > 1. To see why, notice that ln(x + 1) / ln x decreases slowly with x, but ln(x + 2) / ln(x + 1) is evaluated at a larger base and is pulled down more, making q smaller. Verification / Alternative check: We can reason qualitatively: for large x, (x + 1) is only slightly larger than x, so p is close to 1 from above. Meanwhile, x + 2 is only slightly larger than x + 1, so q is also close to 1 but with a slightly larger base in the denominator, which pushes q a bit lower than p. The numerical checks at several integer values confirm this consistent inequality p > q. Why Other Options Are Wrong: p < q: Contradicted by direct substitution for x = 2, 3, 4, all giving p > q. p = q: If they were equal for all x, numeric testing would show equality, but it does not. can't be determined: The pattern is clear and supported both numerically and conceptually, so the relationship is determinable as p > q. Common Pitfalls: A frequent mistake is to treat the increase in argument (from x + 1 to x + 2) as dominant and ignore the effect of increasing the base. Another error is assuming that logs with nearby arguments and bases automatically become equal. It is essential to remember that for base greater than 1, increasing the base while also increasing the argument does not guarantee any specific order without proper analysis. Final Answer: For all natural numbers x > 1, we have p > q.

Question 5

In common (base 10) logarithms, what is the characteristic of log 0.0000134?

Accepted Answer

Introduction / Context: This question focuses on the concept of the characteristic of a common (base 10) logarithm. For numbers less than 1, the characteristic is negative and tells us how far the number is from 1 in terms of powers of 10. Understanding this is essential for reading and using logarithm tables correctly. Given Data / Assumptions: We are working with common logarithms (base 10). The number is 0.0000134. We are asked only for the characteristic, not the mantissa. Concept / Approach: For a positive number N, the common logarithm log₁₀ N can be written as C + M, where C is the characteristic (integer part) and M is the mantissa (fractional part). For N between 0 and 1, the characteristic is negative. A useful rule is: if the first non-zero digit after the decimal occurs in the (k + 1)th place, then the characteristic is -(k + 1). Alternatively, rewrite the number in scientific notation N = a × 10^n, where 1 ≤ a < 10, and then log N = n + log a; here n will be the characteristic if a is between 1 and 10. Step-by-Step Solution: Write 0.0000134 in scientific notation. We can move the decimal point 5 places to the right: 0.0000134 = 1.34 × 10^-5. Then log(0.0000134) = log(1.34 × 10^-5). Using log(ab) = log a + log b, this becomes log 1.34 + log 10^-5. log 10^-5 = -5, and log 1.34 is a positive fraction between 0 and 1. Therefore log(0.0000134) = -5 + (a fractional mantissa). The characteristic is the integer part, which is -5. Verification / Alternative check: We can also count the zeros after the decimal point before the first non-zero digit. In 0.0000134, after the decimal point there are four zeros before the 1. The first significant digit 1 is in the fifth place after the decimal. By the rule, characteristic = -(number of digits to the first significant digit) = -5. This matches the previous method. Why Other Options Are Wrong: 5: This would correspond to a number around 10^5, which is huge compared to 0.0000134, which is very small. 6 and -6: These would indicate the number is closer to 10^6 or 10^-6. But 0.0000134 is equal to 1.34 × 10^-5, so the power of 10 involved is -5, not -6 or 6. Common Pitfalls: Students often miscount the zeros or confuse the sign for numbers less than 1. Another mistake is to think that the characteristic is simply the number of zeros, not the number of digits to the first non-zero digit. Always check by converting to scientific notation and identifying the power of 10 explicitly. Final Answer: The characteristic of log 0.0000134 is -5.

Question 6

Find the value of the sum log (tan 10°) + log (tan 20°) + log (tan 30°) + … + log (tan 80°), where all logarithms are to base 10.

Accepted Answer

Introduction / Context: This trigonometric logarithm problem asks you to evaluate a sum of logarithms of tangent values at specific angles. The angles come in complementary pairs, and the key insight is to use the identity involving tan θ and tan(90° − θ). Understanding this symmetry helps dramatically simplify the expression without computing any actual tangent values. Given Data / Assumptions: The sum is S = log(tan 10°) + log(tan 20°) + log(tan 30°) + log(tan 40°) + log(tan 50°) + log(tan 60°) + log(tan 70°) + log(tan 80°). All logarithms are base 10. Angles are in degrees, and 0° < angle < 90°. Concept / Approach: We make use of the identity tan(90° − θ) = cot θ. The product tan θ × tan(90° − θ) = 1, since tan θ × cot θ = 1. Taking logarithms, log(tan θ) + log(tan(90° − θ)) = log 1 = 0. The given angles can be grouped into complementary pairs: (10°, 80°), (20°, 70°), (30°, 60°), and (40°, 50°). Each pair contributes zero to the sum of logs, so the entire sum collapses to zero. Step-by-Step Solution: Recall that tan(90° − θ) = cot θ and tan θ × cot θ = 1. Therefore tan θ × tan(90° − θ) = 1 for 0° < θ < 90°. Taking logarithms, log(tan θ) + log(tan(90° − θ)) = log 1 = 0. Now group the terms in S into complementary pairs: Pair 1: log(tan 10°) + log(tan 80°) = 0. Pair 2: log(tan 20°) + log(tan 70°) = 0. Pair 3: log(tan 30°) + log(tan 60°) = 0. Pair 4: log(tan 40°) + log(tan 50°) = 0. Add all pairs: S = 0 + 0 + 0 + 0 = 0. Verification / Alternative check: If desired, you can approximate a couple of pairs numerically. For example, tan 30° = 1/√3 and tan 60° = √3, so tan 30° × tan 60° = 1. Therefore log(tan 30°) + log(tan 60°) = log 1 = 0. Similar behaviour holds for (10°, 80°) and other pairs, confirming that every complementary pair contributes zero to the sum. Why Other Options Are Wrong: -1, 1/2, and 1: All these would imply that the product of all the tangent values is something other than 1, but the pairing argument shows that each pair multiplies to 1, and hence the overall product is 1 and the sum of logs must be 0. Common Pitfalls: A typical mistake is attempting to compute or approximate each tangent value directly, which is unnecessary and time consuming. Another pitfall is forgetting the identity tan(90° − θ) = cot θ, which is central to the simplification. Also, some students mis-pair angles or assume an incorrect period for tangent, leading to sign mistakes. Final Answer: The value of the sum of logarithms is 0.

Question 7

Evaluate the logarithmic expression: Find the numerical value of log base 2 of (log base 5 of 625), that is, compute log base 2 (log base 5 625).

Accepted Answer

Introduction: This problem checks your understanding of nested logarithms, where one logarithm expression appears inside another. Correctly evaluating such expressions requires applying basic logarithm rules in the right order and simplifying step by step. Given Data / Assumptions: We must find the value of log base 2 (log base 5 625). All logarithms are assumed to have positive arguments and valid bases (not equal to 1). 625 is a power of 5, which will simplify the inner logarithm. Concept / Approach: To solve nested logarithms, first evaluate the inner logarithm completely, then treat its result as a regular number when evaluating the outer logarithm. Here, the inner term log base 5 of 625 can be simplified by expressing 625 as a power of 5. Once that is done, the outer log base 2 becomes straightforward because we are left with the logarithm of an integer. Step-by-Step Solution: Step 1: Simplify the inner logarithm log base 5 625. Observe that 625 = 5^4. Therefore, log base 5 625 = log base 5 (5^4) = 4. Step 2: Substitute this value into the outer logarithm. The original expression becomes log base 2 (4). Note that 4 = 2^2. Therefore, log base 2 (4) = log base 2 (2^2) = 2. Thus, the value of log base 2 (log base 5 625) is 2. Verification / Alternative check: We can verify by using the definition of a logarithm. Since 5^4 = 625, the inner log is exactly 4. Since 2^2 = 4, the outer log base 2 of 4 is exactly 2. No approximations are involved, so the result is exact and clearly correct. Why Other Options Are Wrong: The options 5, 10, and 15 would require either 625 or 4 to be a very different power of 5 or 2, which is not true. For example, log base 2 (4) is definitely not 5 or 10. These incorrect values usually arise from confusing the base and argument or from skipping steps when simplifying powers. Common Pitfalls: Common mistakes include trying to directly combine the logs without first evaluating the inner one, or thinking that log base 2 (log base 5 625) somehow equals log base 10 of 625. Remember that the base of a logarithm matters, and each log must be processed with its own base. Always simplify from the inside out when functions are nested. Final Answer: The value of log base 2 (log base 5 625) is 2.

Question 8

Let a, b, and c be respectively the p-th, q-th, and r-th terms of a geometric progression (G.P.). Evaluate the logarithmic expression: (q − r) log a + (r − p) log b + (p − q) log c.

Accepted Answer

Introduction: This question tests your understanding of geometric progressions (G.P.) and logarithm properties. You are asked to simplify an algebraic expression involving logs of three terms of a G.P. with indices p, q, and r. The elegance of this problem lies in recognizing the common ratio structure and using logarithm rules to show that the whole expression simplifies neatly. Given Data / Assumptions: a is the p-th term of a G.P. b is the q-th term of the same G.P. c is the r-th term of the same G.P. We must evaluate (q − r) log a + (r − p) log b + (p − q) log c. We assume the base of the logarithm is fixed and positive, not equal to 1. Concept / Approach: In a geometric progression, each term can be written in terms of the first term and the common ratio. If the first term is A and the common ratio is R, then the n-th term is A * R^(n−1). Taking logs converts products into sums and powers into coefficients. By expressing log a, log b, and log c in terms of log A and log R, we can substitute into the given expression and see large cancellations. Step-by-Step Solution: Let the first term of the G.P. be A and the common ratio be R. Then a = A * R^(p−1), b = A * R^(q−1), c = A * R^(r−1). Take logarithms: log a = log A + (p−1) log R log b = log A + (q−1) log R log c = log A + (r−1) log R Now form the expression E = (q−r) log a + (r−p) log b + (p−q) log c. Substitute logs: E = (q−r)[log A + (p−1) log R] + (r−p)[log A + (q−1) log R] + (p−q)[log A + (r−1) log R] Group coefficients of log A and log R separately. Coefficient of log A: (q−r) + (r−p) + (p−q) = 0. Coefficient of log R: (q−r)(p−1) + (r−p)(q−1) + (p−q)(r−1). This second coefficient also simplifies to 0 after expansion and cancellation. Hence E = 0 * log A + 0 * log R = 0. Verification / Alternative check: You can test with a concrete example. Take a simple G.P. like 2, 4, 8, 16, ... with A = 2 and R = 2, and choose some values for p, q, r. Compute a, b, c and evaluate the expression numerically with logs. You will always obtain zero, confirming the algebraic result. Why Other Options Are Wrong: The values 1, −1, or pqr introduce nonzero constants or dependence on indices that do not appear after simplification. Because both log A and log R cancel out entirely, the expression cannot depend on p, q, r, or any fixed nonzero constant; it must be exactly zero. Common Pitfalls: Many learners forget to write the G.P. terms in the general form A * R^(n−1), or they make sign mistakes when expanding the coefficients. Others attempt to treat log a, log b, and log c as unrelated, which hides the underlying G.P. structure. Always express all related quantities in a common parametric form before simplifying. Final Answer: The expression simplifies to 0 for any geometric progression and any integers p, q, and r.

Question 9

Evaluate the logarithmic expression: Find the exact numerical value of log base 343 of 7, that is, compute log base 343 (7).

Accepted Answer

Introduction: This question focuses on evaluating a simple logarithm where the base and the argument are powers of the same number. Recognizing this power relationship allows you to compute the logarithm exactly without using calculators or approximate values. Given Data / Assumptions: We must find log base 343 (7). 343 is a composite number, and we should try to express it in terms of a simpler base. The logarithm is taken with a valid base greater than 0 and not equal to 1. Concept / Approach: When both the base and the argument of a logarithm are powers of the same number, we can write them as a^m and a^n, respectively. Then log base a^m (a^n) equals n / m. This comes directly from the definition of a logarithm and properties of exponents. Here, both 343 and 7 relate naturally to the base 7. Step-by-Step Solution: Step 1: Express 343 as a power of 7. 343 = 7 × 7 × 7 = 7^3. So the base 343 is 7^3, and the argument is 7^1. Step 2: Rewrite the logarithm using these powers. log base 343 (7) = log base 7^3 (7^1). Step 3: Use the identity log base a^m (a^n) = n / m. Here a = 7, m = 3, and n = 1. Therefore, log base 7^3 (7^1) = 1 / 3. So log base 343 (7) = 1/3. Verification / Alternative check: By definition, log base 343 (7) = x means 343^x = 7. If x = 1/3, then 343^(1/3) is the cube root of 343, which is 7. Since this matches the argument, x = 1/3 is indeed the correct logarithm value. Why Other Options Are Wrong: The value 3 would imply 343^3 = 7, which is false because 343^3 is much larger than 7. The values −3 and −1/3 would imply 343 raised to a negative power equals 7, which would yield a fraction less than 1, not 7. Therefore, none of these can satisfy the logarithm equation. Common Pitfalls: Learners sometimes reverse the roles of base and argument, accidentally evaluating log base 7 (343) instead of log base 343 (7). Others forget the exponent rule and try to approximate using decimal logs. Always rewrite both numbers as powers of the same base and apply the n / m rule carefully. Final Answer: The exact value of log base 343 (7) is 1/3.

Question 10

Solve the linear equation in one variable: Given 5x − 17 = −x + 7, find the value of x that satisfies this equation.

Accepted Answer

Introduction: This problem is a straightforward example of solving a linear equation in one variable. Such equations are a foundation for algebra and appear frequently in aptitude and entrance examinations. The goal is to isolate x on one side of the equation using basic algebraic operations. Given Data / Assumptions: Equation: 5x − 17 = −x + 7 x is a real number. We must find the unique value of x that makes both sides of the equation equal. Concept / Approach: To solve a linear equation, we typically collect all x terms on one side and all constant terms on the other side. Then we combine like terms and divide to isolate x. The operations used must be valid on both sides of the equation to maintain equality. Because this is a first degree equation, we expect exactly one solution. Step-by-Step Solution: Start with: 5x − 17 = −x + 7 Step 1: Move all x terms to the left side by adding x to both sides. 5x − 17 + x = −x + 7 + x This gives: 6x − 17 = 7 Step 2: Move the constant term −17 to the right side by adding 17 to both sides. 6x − 17 + 17 = 7 + 17 This gives: 6x = 24 Step 3: Divide both sides by 6 to isolate x. x = 24 / 6 = 4 Therefore, x = 4 is the solution. Verification / Alternative check: Substitute x = 4 back into the original equation. Left side: 5(4) − 17 = 20 − 17 = 3. Right side: −4 + 7 = 3. Both sides equal 3, so x = 4 satisfies the equation exactly. No other value will do so for a linear equation of this type. Why Other Options Are Wrong: If you substitute x = 5, x = 7, or x = 9 into the equation, the left and right sides will not match. For example, for x = 5, left side is 25 − 17 = 8, while right side is −5 + 7 = 2. Hence these values cannot be solutions. Common Pitfalls: Typical mistakes include sign errors when moving terms across the equal sign, or incorrect arithmetic when adding or dividing. Some students also try to divide too early without first combining like terms. Proceeding step by step and checking each operation avoids these errors. Final Answer: The value of x that satisfies the equation is 4.

Question 11

Let A = log base 32 of 1875 and B = log base 243 of 2187. Based on these definitions, which of the following relationships between A and B is correct?

Accepted Answer

Introduction: This question tests your ability to compare two logarithmic expressions without necessarily computing them to many decimal places. By recognizing the structure of the numbers involved, you can simplify or estimate each logarithm and then decide whether A is less than, equal to, or greater than B. Given Data / Assumptions: A = log base 32 (1875) B = log base 243 (2187) We must determine whether A < B, A = B, A > B, or if the relationship cannot be determined. Concept / Approach: The key idea is to exploit the fact that 32 and 243 are perfect powers and that 2187 is also a perfect power. Specifically, 32 = 2^5, 243 = 3^5, and 2187 = 3^7. For B, this allows an exact and simple calculation. For A, we can express 1875 in terms of prime factors and then obtain a reasonable estimate by comparing exponents, or by using change of base thinking qualitatively. Step-by-Step Solution: Step 1: Compute B exactly. 243 = 3^5 and 2187 = 3^7. Thus B = log base 3^5 (3^7) = 7 / 5 = 1.4. Step 2: Analyze A = log base 32 (1875). We know 32 = 2^5. Factor 1875: 1875 = 3 × 5^4. So A = log base 2^5 (3 × 5^4). This does not simplify to a simple rational like B, but we can compare orders of magnitude. Note that 2^5 = 32 and 32^2 = 1024, 32^3 = 32768. 1875 lies between 32^1 and 32^2, but much closer to 32^2 than to 32. As a result, log base 32 (1875) is between 1 and 2, and closer to 2 than to 1. A rough estimation shows A is a bit above 2, while B is exactly 1.4. Therefore, A is greater than B. Verification / Alternative check: Using approximate common logarithms, log10(1875) is about 3.27 and log10(32) is about 1.51. Then A ≈ 3.27 / 1.51 ≈ 2.16. Since B = 1.4 exactly, this confirms that A > B, with a comfortable margin so that rounding errors cannot change the ordering. Why Other Options Are Wrong: A < B would require A to be less than 1.4, which contradicts the fact that 1875 is far larger than 32 and close to 32^2. A = B is impossible because we have computed B exactly and A approximately as a clearly larger number. Saying the relationship cannot be determined is incorrect because we have enough information to compare them. Common Pitfalls: Some learners attempt to compare the arguments 1875 and 2187 directly, ignoring the different bases. Others may forget that larger bases shrink logarithm values and smaller bases increase them. Always rewrite numbers in terms of prime powers where possible and use the definition of logarithms logically rather than guessing. Final Answer: We conclude that A is greater than B, so A > B is correct.

Question 12

If log 2 = 0.3010 and log 3 = 0.4771, the values of log5 512 is

Accepted Answer

ANS: log5512 = log512/log5 = log 2 9 log 10 / 2 = 9 log 2 log 10 - log 2 = 9 * 0 . 3010 1 - 0 . 3010 =2.709/0.699 =2709/699 =3.876

Question 13

If log 27 = 1.431, then the value of log 9 is

Accepted Answer

log 27 = 1.431 log 3 3 = 1.431 3 log 3 = 1.431 log 3 = 0.477 log 9 = log( 3 2 )= 2 log 3 = (2 x 0.477) = 0.954

Question 14

If log 2 = 0.30103, Find the number of digits in 256 is

Accepted Answer

log ( 2 56 ) =56*0.30103 =16.85768. Its characteristics is 16. Hence, the number of digits in 2 56 is 17.

Question 15

If logaab = x, then logbab = ?

Accepted Answer

x/(x-1)

Question 16

If logx916 = -12, then the value of x?

Accepted Answer

256/81

Question 17

If log72 = m, then log4928 is equal to ?

Accepted Answer

log 49 28 = 1 2 log 7 7 × 4 = 1 2 + 1 2 2 log 7 2 = 1 2 + log 7 2 = 1 2 + m = 1 + 2 m 2 .

Question 18

Find the logarithm of 144 to the base 23 :

Accepted Answer

log 2 3 144 = 4

Question 19

If logxl+m-2n = logym+n-2l = logzn+l-2m , then xyz is equal to

Accepted Answer

1

Question 20

If log 64 = 1.8061, then the value of log 16 will be (approx)?

Accepted Answer

Given that, log 64 = 1.8061 i.e log 4 3 = 1 . 8061 --> 3 log 4 = 1.8061 --> log 4 = 0.6020 --> 2 log 4 = 1.2040 ⇒ log 4 2 = 1 . 2040 Therefore, log 16 = 1.2040

Logarithm Questions