Question 1

If $\log_{10} 2 = 0.3010$, the value of $\log_{10} 80$ is

Accepted Answer

Concept & Formula: To find the logarithm of a composite number, you must break the number down into its prime factors, particularly those related to the given values and the logarithm's base. Here, we will decompose 80 into factors of 2 and 10, and then apply the Product Rule and Power Rule of logarithms. 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 value: \log_{10} 2 = 0.3010 Step 1: Express 80 using base 10 and factors of 2. Since 80 = 8 × 10, and 8 = 2^3, we can write: 80 = 2^3 × 10 Step 2: Apply the logarithm to both sides and use the Product Rule to separate the terms. \log_{10} 80 = \log_{10} (2^3 × 10) \log_{10} 80 = \log_{10} (2^3) + \log_{10} 10 Step 3: Apply the Power Rule to the first term, bringing the exponent to the front. \log_{10} 80 = 3 \log_{10} 2 + \log_{10} 10 Step 4: Substitute the known values. We are given \log_{10} 2 = 0.3010, and we know intrinsically that \log_{10} 10 = 1. \log_{10} 80 = 3(0.3010) + 1 Step 5: Multiply and add to get the final result. \log_{10} 80 = 0.9030 + 1 \log_{10} 80 = 1.9030 Exam Strategy & Shortcut: Characteristic and Mantissa Shortcut: Recognize that 80 = 8 × 10^1. The "× 10^1" guarantees the answer will simply be the log of 8 plus exactly 1 (the characteristic). Mentally calculate \log 8 = 3 × \log 2 = 3 × 0.3010 = 0.9030. Add 1 to the whole number place to account for the single zero in 80, yielding 1.9030 instantly. Common Pitfall: A frequent trap is breaking 80 down incorrectly or needlessly into smaller primes without utilizing the base (10). For instance, factoring 80 as 2^4 × 5 requires you to calculate both \log 2 and \log 5, which introduces an extra step and a higher chance of arithmetic error compared to keeping the factor of 10 intact. Final Answer: Therefore, the correct answer is 1.9030.

Question 2

If $\log 3 = 0.477$ and $(1000)^x = 3$, then $x$ equals

Accepted Answer

Concept & Formula: The core concept is to solve for an unknown exponent by applying logarithms to both sides of an exponential equation. This allows us to use the power rule to bring the exponent down as a multiplier. Formula used: Power Rule: \log(a^b) = b \log a Base 10 Identity: \log_{10} 1000 = \log_{10} (10^3) = 3 Step-by-Step Solution: Given the equation: (1000)^x = 3 Step 1: Take the common logarithm (base 10) of both sides. \log((1000)^x) = \log 3 Step 2: Apply the power rule to bring x to the front. x \log 1000 = \log 3 Step 3: Evaluate \log 1000. Since 1000 = 10^3, its base-10 logarithm is 3. Substitute this and the given value for \log 3 (0.477). x(3) = 0.477 Step 4: Solve for x by dividing both sides by 3. x = (0.477/3) x = 0.159 Exam Strategy & Shortcut: Mental Exponent Conversion: Recognize immediately that 1000 is 10^3. Therefore, (1000)^x is simply 10^{3x}. If 10^{3x} = 3, converting this to logarithmic form directly yields 3x = \log 3. You can instantly bypass the intermediate steps, plug in 0.477, and mentally divide by 3 to get 0.159. Common Pitfall: A common mistake is attempting to divide the logarithm arguments directly or confusing the base. Some students might incorrectly assume x = (3/1000) or mishandle the base 10 properties, leading to answers off by decimal places, like option (a) or (b). Final Answer: Therefore, the correct answer is 0.159.

Question 3

If $\log_{10} 20 = 1.3010$ and $\log_{10} 30 = 1.4771$, then $\log_{10} (60000)$ is equal to

Accepted Answer

Concept & Formula: This question assesses your ability to factor a large number into the specific components provided in the prompt. By factoring 60000 using 20 and 30, we can utilize the Product Rule of logarithms. Formula used: Product Rule: \log(abc) = \log a + \log b + \log c Step-by-Step Solution: Given the values: \log_{10} 20 = 1.3010 \log_{10} 30 = 1.4771 Step 1: Break down the number 60000 into factors that include 20, 30, and powers of 10. Notice that 20 × 30 = 600. To get 60000, we need to multiply 600 by 100. 60000 = 20 × 30 × 100 Step 2: Apply the logarithm to both sides and use the Product Rule to expand the expression. \log_{10} 60000 = \log_{10} (20 × 30 × 100) \log_{10} 60000 = \log_{10} 20 + \log_{10} 30 + \log_{10} 100 Step 3: Substitute the known values and evaluate \log_{10} 100. (Since 100 = 10^2, \log_{10} 100 = 2). \log_{10} 60000 = 1.3010 + 1.4771 + 2 Step 4: Add the numbers together. \log_{10} 60000 = 2.7781 + 2 \log_{10} 60000 = 4.7781 Exam Strategy & Shortcut: Direct Substitution via Scientific Notation: You can also break 60000 into 6 × 10^4. The log of 6 × 10^4 is simply \log 6 + 4. How to find \log 6? The problem gave \log 20 = \log (2 × 10) = \log 2 + 1 = 1.3010, meaning \log 2 = 0.3010. Similarly, \log 30 = \log 3 + 1 = 1.4771, meaning \log 3 = 0.4771. \log 6 = \log 2 + \log 3 = 0.3010 + 0.4771 = 0.7781. Add the 4 from the 10^4, and you get 4.7781. Both methods are equally valid and fast. Common Pitfall: A common error is to over-factor the numbers into pure primes (like 2 and 3) without noticing that the given values (\log 20 and \log 30) can be used directly as larger "building blocks", saving calculation time. Final Answer: Therefore, the correct answer is 4.7781.

Question 4

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 5

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 6

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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