Question 1

If $\log 2 = 0.30103$, the number of digits in $2^{64}$ is

Accepted Answer

Concept & Formula: The number of digits in an integer N can be directly determined using its common logarithm (base 10). When you calculate \log_{10}(N), the whole number part of the result (the characteristic) determines the digits. Formula: Number of digits = Characteristic + 1. Step-by-Step Solution: Given the value: \log 2 = 0.30103 Step 1: Take the common logarithm of the target number, 2^{64}. Let x = 2^{64} \log x = \log (2^{64}) Step 2: Apply the Power Rule of logarithms to bring the exponent to the front. \log (2^{64}) = 64 × \log 2 Step 3: Substitute the known value of \log 2 and multiply. 64 × 0.30103 = 19.26592 Step 4: Identify the characteristic and calculate the number of digits. The integral part (characteristic) of 19.26592 is 19. Number of digits = 19 + 1 = 20 Exam Strategy & Shortcut: Approximation Method: You don't need the exact decimal value to find the characteristic. Round \log 2 to 0.3 for a quick mental check. 64 × 0.3 = 19.2 The whole number is 19. Add 1 to get 20. Even if the actual value was slightly higher, it would not cross into the 20.x range, making this approximation incredibly safe and fast for competitive exams. Common Pitfall: Students often forget the final step of adding 1 to the characteristic. They correctly calculate 19.26, identify 19, and mistakenly select 19 as the final answer. Always remember: Digits = Characteristic + 1. Final Answer: Therefore, the correct answer is 20.

Question 2

If $\log 2 = 0.30103$, the number of digits in $4^{50}$ is

Accepted Answer

Concept & Formula: To find the number of digits in an exponential number, we use its base-10 logarithm. The number of digits is always one more than the characteristic (the integer part) of the logarithm. Before calculating, if the base of the exponent (in this case, 4) can be simplified into a prime base we know (like 2), we should apply exponent rules first. Formula: Number of digits = Characteristic + 1 Step-by-Step Solution: Given the value: \log 2 = 0.30103 Step 1: Simplify the target expression 4^{50} by converting the base 4 into base 2. 4 = 2^2 4^{50} = (2^2)^{50} Step 2: Multiply the exponents. (2^2)^{50} = 2^{100} Step 3: Take the common logarithm of the simplified expression and apply the Power Rule. \log (2^{100}) = 100 × \log 2 Step 4: Substitute the given value for \log 2. 100 × 0.30103 = 30.103 Step 5: Determine the number of digits. The characteristic (integer part) is 30. Number of digits = 30 + 1 = 31 Exam Strategy & Shortcut: Zero-Math Shift: Recognizing 4^{50} as 2^{100} is the ultimate shortcut here. Multiplying any decimal by 100 simply shifts the decimal point two places to the right. You instantly transform 0.30103 into 30.103. Grab the 30, add 1, and mark 31 as your answer in under 5 seconds without touching a pen. Common Pitfall: A less efficient method is to find \log 4 first (2 × 0.30103 = 0.60206) and then multiply by 50. While mathematically sound, multiplying a long decimal by 50 introduces more room for simple arithmetic errors compared to simply shifting a decimal point with 100. Final Answer: Therefore, the correct answer is 31.

Question 3

If $\log 2 = 0.30103$, then the number of digits in $5^{20}$ is

Accepted Answer

Concept & Formula: To calculate the number of digits for a power of 5 when only given \log 2, we must utilize the base-10 complement rule. Because 10 = 2 × 5, their logarithmic values are intimately linked. Formulas used: 1. Complement Rule: \log 5 = \log(10/2) = 1 - \log 2 2. Number of Digits = Characteristic of \log(N) + 1 Step-by-Step Solution: Given the value: \log 2 = 0.30103 Step 1: First, we need to find the value of \log 5. \log 5 = 1 - \log 2 \log 5 = 1 - 0.30103 = 0.69897 Step 2: Set up the logarithm for the target expression, 5^{20}, and use the Power Rule. \log (5^{20}) = 20 × \log 5 Step 3: Substitute the value of \log 5 we just found and multiply. 20 × 0.69897 = 13.9794 Step 4: Find the number of digits by taking the characteristic and adding 1. The characteristic here is 13. Number of digits = 13 + 1 = 14 Exam Strategy & Shortcut: Strategic Multiplication & Approximation: You don't need to calculate 20 × 0.69897 to five decimal places. You know \log 5 is roughly 0.7. Calculate 20 × 0.7 = 14. Since \log 5 (0.69897) is actually *slightly less* than 0.7, the product 20 × 0.69897 will be *slightly less* than 14 (specifically, in the 13.9 range). Therefore, the characteristic is definitively 13. Add 1 to get 14 digits. Common Pitfall: The most common mistake is failing to recognize how to extract \log 5 from \log 2. Students without a calculator might get stuck here if they don't remember the 1 - \log 2 identity, guessing the answer blindly instead. Final Answer: Therefore, the correct answer is 14.

Question 4

If $\log_{10} a = p$ and $\log_{10} b = q$, then what is $\log_{10} (a^p b^q)$ equal to?

Accepted Answer

Concept & Formula: This question tests the ability to fully expand a logarithmic expression using fundamental properties and substitute given algebraic values. Formulas used: 1. Product Rule: \log(xy) = \log x + \log y 2. Power Rule: \log(x^n) = n \log x Step-by-Step Solution: Given the expression: \log_{10} (a^p b^q) Step 1: Apply the logarithmic Product Rule to separate the variables being multiplied inside the argument. \log_{10} (a^p) + \log_{10} (b^q) Step 2: Apply the logarithmic Power Rule to bring the exponents (p and q) to the front of their respective terms as coefficients. p \log_{10} a + q \log_{10} b Step 3: Substitute the known values provided in the question (\log_{10} a = p and \log_{10} b = q). p(p) + q(q) Step 4: Multiply the terms together to get the final algebraic expression. p^2 + q^2 Exam Strategy & Shortcut: Direct Visual Mapping: This is a pure property-based question that should require zero calculation time. Read the exponents as coefficients immediately in your head: a^p gives p × \log a (which maps to p × p). b^q gives q × \log b (which maps to q × q). Because it is a product (a^p × b^q), you add the two results. The answer p^2 + q^2 should be visually apparent without writing down a single intermediate step. Common Pitfall: A severe mistake is misinterpreting the product rule and assuming that \log(a^p b^q) equals (\log a^p) × (\log b^q). This incorrect multiplication of terms leads directly to the trap option (c) p^2q^2. Remember: The log of a product is the *sum* of the individual logs. Final Answer: Therefore, the correct answer is p^2 + q^2.

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