Question 1

If $\log \frac{a}{b} + \log \frac{b}{a} = \log (a + b)$, then

Accepted Answer

Concept & Formula: This problem requires the application of the product rule for logarithms to combine the terms on the left side of the equation. \log x + \log y = \log(xy) Additionally, you must recognize the fundamental logarithm identity that the log of 1 is always 0. \log 1 = 0 Step-by-Step Solution: • Given: An equation involving the sum of two logarithmic fractions. \log (a/b) + \log (b/a) = \log(a + b) • Calculation: Apply the product rule to combine the two logarithms on the left side into a single logarithm. \log ( (a/b) × (b/a) ) = \log(a + b) • Simplify the argument inside the combined logarithm. The numerator and denominator cancel each other out entirely. \log(1) = \log(a + b) • Since the logarithm of 1 to any base is always 0, the left side becomes 0. 0 = \log(a + b) • Assuming the standard base 10 (or any valid base x), convert the logarithmic equation into an exponential equation to solve for a + b. 10^0 = a + b 1 = a + b Exam Strategy & Shortcut: Notice immediately that (a/b) and (b/a) are reciprocals. When you add their logarithms, you are multiplying reciprocals inside a single log, which always yields \log(1) = 0. Therefore, you can jump straight to 0 = \log(a+b), which instantly means a+b = 1. This mental shortcut saves crucial exam time. Common Pitfall: A common mistake is incorrectly expanding \log(a+b) on the right side into \log a + \log b, which is mathematically invalid. Never confuse the product rule \log(ab) = \log a + \log b with an additive argument \log(a+b). Treat \log(a+b) as a single unbreakable entity until you convert it to exponential form. Final Answer: Therefore, the correct answer is a + b = 1.

Question 2

$\left[ \log \left( \frac{a^2}{bc} \right) + \log \left( \frac{b^2}{ac} \right) + \log \left( \frac{c^2}{ab} \right) \right]$ is equal to

Accepted Answer

Concept & Formula: This expression is simplified using the logarithmic product rule. The sum of multiple logarithms with the same base is equal to the logarithm of the product of their arguments. \log x + \log y + \log z = \log(xyz) Step-by-Step Solution: • Given: A summation of three logarithmic terms containing algebraic fractions. \log ( (a^2/bc) ) + \log ( (b^2/ac) ) + \log ( (c^2/ab) ) • Calculation: Apply the product rule to merge the three logarithms into a single logarithmic expression. Multiply the three fractional arguments together. \log ( (a^2/bc) × (b^2/ac) × (c^2/ab) ) • Combine the numerators and denominators. \log ( (a^2 \cdot b^2 \cdot c^2/(bc) \cdot (ac) \cdot (ab)) ) • Rearrange and simplify the denominator by multiplying the common variables. a \cdot a = a^2, b \cdot b = b^2, and c \cdot c = c^2. \log ( (a^2 b^2 c^2/a^2 b^2 c^2) ) • The numerator and denominator are identical, so the fraction simplifies to 1. \log(1) • The logarithm of 1, regardless of the base, is always 0. 0 Exam Strategy & Shortcut: When faced with cyclical symmetric expressions like (a^2/bc), (b^2/ac), and (c^2/ab) being added together inside logs, you should suspect they will cancel out. A quick mental check multiplying the tops (a^2b^2c^2) and bottoms (a^2b^2c^2) confirms they reduce to 1. Since \log(1) = 0, you can select option (a) in under 10 seconds without writing anything down. Common Pitfall: Some students attempt to apply the quotient rule to expand each term individually (e.g., \log(a^2) - \log(bc)). While mathematically correct, it creates a massive, sprawling expression that is highly prone to sign errors and takes significantly longer to resolve. Stick to the product rule for efficiency. Final Answer: Therefore, the correct answer is 0.

Question 3

$\frac{1}{\log_a b} 	imes \frac{1}{\log_b c} 	imes \frac{1}{\log_c a}$ is equal to

Accepted Answer

Concept & Formula: This problem requires two core properties of logarithms: the Reciprocal Rule (a derivation of the base-change formula) and the Chain Rule. (1/\log_x y) = \log_y x \log_x y × \log_y z = \log_x z Step-by-Step Solution: • Given: A product of three reciprocal logarithms with varying bases. (1/\log_a b) × (1/\log_b c) × (1/\log_c a) • Calculation: First, apply the reciprocal rule to move the logarithms from the denominator to the numerator. This swaps the base and the argument for each term. \log_b a × \log_c b × \log_a c • Rearrange the terms to make the chain rule visually obvious. Place them so the argument of one log matches the base of the next. \log_c b × \log_b a × \log_a c • Apply the logarithmic chain rule to the first two terms (\log_c b × \log_b a). The b's cancel out, collapsing the terms. \log_c a × \log_a c • Apply the chain rule again to the remaining terms. The a's cancel out. \log_c c • Apply the identity rule for logarithms, where a base to the power of 1 equals itself (\log_x x = 1). 1 Exam Strategy & Shortcut: An alternative and often faster visualization uses the standard Base Change Formula: \log_x y = (\log y/\log x). Applying this to the reciprocal format means (1/\log_a b) = (\log a/\log b). Transforming all three terms gives: (\log a/\log b) × (\log b/\log c) × (\log c/\log a). You can immediately see that every numerator perfectly cancels every denominator, leaving exactly 1. Common Pitfall: A major trap is confusing multiplication with addition. Students sometimes mistakenly try to convert the product of these fractions into the logarithm of a sum, which is mathematically impossible. Always rely on the base change formula when multiplying logarithms with different bases. Final Answer: Therefore, the correct answer is 1.

Question 4

$\left[ \frac{1}{(\log_a bc) + 1} + \frac{1}{(\log_b ca) + 1} + \frac{1}{(\log_c ab) + 1} \right]$ is equal to

Accepted Answer

Concept & Formula: This problem is solved by rewriting the constant 1 as a logarithm with a matching base to combine terms, and then utilizing the logarithmic reciprocal rule. \log_x x = 1 (1/\log_x y) = \log_y x Step-by-Step Solution: • Given: We need to evaluate the sum of three algebraic fractions involving logarithms. (1/\log_a bc + 1) + (1/\log_b ca + 1) + (1/\log_c ab + 1) • Calculation: To combine the terms in the denominators, replace the 1 in each fraction with a logarithm that matches the base of the adjacent logarithm. For the first term, 1 = \log_a a. For the second term, 1 = \log_b b. For the third term, 1 = \log_c c. (1/\log_a bc + \log_a a) + (1/\log_b ca + \log_b b) + (1/\log_c ab + \log_c c) • Apply the product rule (\log x + \log y = \log(xy)) to merge the denominators. (1/\log_a(abc)) + (1/\log_b(abc)) + (1/\log_c(abc)) • Now, use the reciprocal rule to move the logarithms to the numerators. This swaps the base and the argument. Notice how this perfectly aligns all three bases to (abc). \log_{abc} a + \log_{abc} b + \log_{abc} c • Apply the product rule once more to combine this sum into a single logarithm. \log_{abc}(a × b × c) \log_{abc}(abc) • Apply the identity rule. Since the base and argument are identical, the result is 1. 1 Exam Strategy & Shortcut: When you encounter symmetric, cyclical expressions in algebra or logarithms (where terms rotate uniformly like a \rightarrow b \rightarrow c), the expressions almost always collapse to 0, 1, or a simple product. Recognizing that the denominators will all normalize to \log(abc) allows you to mentally flip them to base abc and add them up to 1 in just a few seconds without writing out the intermediate steps. Common Pitfall: Students frequently get stuck because they do not realize the constant 1 can be flexibly rewritten as \log_a a, \log_b b, etc. Leaving the 1 as a standard integer makes it impossible to combine the terms in the denominator. Always look for ways to unify formats. Final Answer: Therefore, the correct answer is 1.

Question 5

The value of $\left[ \frac{1}{\log_{(p/q)} x} + \frac{1}{\log_{(q/r)} x} + \frac{1}{\log_{(r/p)} x} \right]$ is

Accepted Answer

Concept & Formula: This question tests your ability to use the base change reciprocal property to convert fractions into manageable additive logarithms. (1/\log_b a) = \log_a b \log_x a + \log_x b = \log_x(ab) Step-by-Step Solution: • Given: An expression with the variable x as the argument in the denominators, and fractions as the bases. (1/\log_{(p/q)) x} + (1/\log_{(q/r)) x} + (1/\log_{(r/p)) x} • Calculation: Use the reciprocal rule to bring each logarithm up to the numerator. This swaps the fractional base with the argument x, giving all three logarithms a common base of x. \log_x ((p/q)) + \log_x ((q/r)) + \log_x ((r/p)) • Since all logarithms now share the same base x and are being added together, apply the product rule to combine them. Multiply the arguments together inside a single logarithm. \log_x ( (p/q) × (q/r) × (r/p) ) • Simplify the product inside the argument. The p in the numerator cancels the p in the denominator, the q cancels the q, and the r cancels the r. \log_x (1) • The logarithm of 1 to any valid base is exactly 0. 0 Exam Strategy & Shortcut: Identify the common structure: you have reciprocal logs that all share the same argument x. This means they will all instantly convert to logs with base x. Because it's an addition chain, you just need to multiply the original bases together. Visually cross-canceling (p/q) × (q/r) × (r/p) yields 1 in your head. Since \log_x(1) = 0, you can mark the answer in under 10 seconds. Common Pitfall: A major misstep is trying to use the quotient rule to expand the bases in the denominator (e.g., trying to turn \log_{(p/q)} x into a subtraction). This creates complex nested fractions that are incredibly difficult to resolve. Always look to unify the bases first, usually via the reciprocal rule. Final Answer: Therefore, the correct answer is 0.

Question 6

If $\log_{10} 7 = a$, then $\log_{10} \left( \frac{1}{70} \right)$ is equal to

Accepted Answer

Concept & Formula: This problem requires applying the quotient rule and product rule of logarithms to break down the given expression, alongside the fundamental identity that the logarithm of the base itself is 1. \log ((x/y)) = \log x - \log y \log(xy) = \log x + \log y Step-by-Step Solution: • Given: We know that \log_{10} 7 = a. We need to evaluate \log_{10} ( (1/70) ). • Calculation: First, apply the quotient rule to expand the target expression. \log_{10} ( (1/70) ) = \log_{10} 1 - \log_{10} 70 • The logarithm of 1 to any base is always 0 (\log_{10} 1 = 0). 0 - \log_{10} 70 = -\log_{10} 70 • Now, break down 70 into factors that include the given value (7) and the base (10), which is 7 × 10. Apply the product rule. - (\log_{10} (7 × 10)) - (\log_{10} 7 + \log_{10} 10) • Substitute the given value \log_{10} 7 = a, and use the identity \log_{10} 10 = 1. - (a + 1) Exam Strategy & Shortcut: Instead of expanding via the quotient rule first, you can use negative exponents: (1/70) = 70^{-1}. Then, \log_{10}(70^{-1}) = -\log_{10} 70. Split 70 into 7 × 10. -\log_{10}(7 × 10) = -(\log_{10} 7 + \log_{10} 10) = -(a + 1). This gets you to the answer in fewer mental steps. Common Pitfall: Students often incorrectly handle the negative sign when expanding - \log_{10}(7 × 10), forgetting to distribute it to both terms, which leads to the wrong answer -a + 1 instead of -(a + 1). Always use parentheses when applying logarithmic rules after a negative sign. Final Answer: Therefore, the correct answer is -(1 + a).

Question 7

If $a = b^x$, $b = c^y$ and $c = a^z$, then the value of $xyz$ is equal to

Accepted Answer

Concept & Strategy: This cyclic equation problem can be solved either by continuous algebraic substitution of the bases or by converting the exponential equations into logarithmic forms and multiplying them together. The substitution method is usually faster. Power of a power rule: (x^m)^n = x^{mn} Step-by-Step Solution: • Given: Three interlocking exponential equations: a = b^x, b = c^y, and c = a^z. • Calculation (Substitution Method): Start with the first equation. a = b^x • Substitute the value of b from the second equation (b = c^y) into the first equation. a = (c^y)^x = c^{xy} • Now, substitute the value of c from the third equation (c = a^z) into the new equation. a = (a^z)^{xy} • Apply the power rule to multiply the exponents. a^1 = a^{xyz} • Since the bases on both sides of the equation are the same (a), their exponents must be equal. 1 = xyz Exam Strategy & Shortcut: An alternative rapid method relies on logarithms. Convert each equation: x = (\log a/\log b) y = (\log b/\log c) z = (\log c/\log a) When you multiply xyz, all the numerators and denominators perfectly cross-cancel: ((\log a/\log b)) × ((\log b/\log c)) × ((\log c/\log a)) = 1. This mental cross-cancellation allows you to see the answer instantly. Common Pitfall: When substituting, students sometimes mistakenly add the exponents instead of multiplying them (e.g., changing (c^y)^x to c^{y+x}). Remember that an exponent raised to another exponent means you multiply the powers. Final Answer: Therefore, the correct answer is 1.

Question 8

If $a = b^2 = c^3 = d^4$, then the value of $\log_a (abcd)$ would be

Accepted Answer

Concept & Strategy: The strategy is to express all variables in terms of a single variable (in this case, a, since it is the base of the logarithm) and then use the logarithmic product rule to expand the expression. \log_x(p \cdot q \cdot r \cdot s) = \log_x p + \log_x q + \log_x r + \log_x s Step-by-Step Solution: • Given: The equality chain a = b^2 = c^3 = d^4. We need to find \log_a (abcd). • Calculation: First, express b, c, and d in terms of a using fractional exponents. b^2 = a \Rightarrow b = a^{(1/2)} c^3 = a \Rightarrow c = a^{(1/3)} d^4 = a \Rightarrow d = a^{(1/4)} • Expand the target logarithmic expression using the product rule. \log_a(abcd) = \log_a a + \log_a b + \log_a c + \log_a d • Substitute the expressions for b, c, and d that we found in terms of a. \log_a a + \log_a(a^{(1/2)}) + \log_a(a^{(1/3)}) + \log_a(a^{(1/4)}) • Use the power rule (\log_x(y^n) = n \log_x y) to bring the fractional exponents to the front. Also, note that \log_a a = 1. 1 + (1/2)\log_a a + (1/3)\log_a a + (1/4)\log_a a • Since \log_a a = 1 in every term, it simplifies cleanly to fractions. 1 + (1/2)(1) + (1/3)(1) + (1/4)(1) 1 + (1/2) + (1/3) + (1/4) Exam Strategy & Shortcut: An even faster method skips expanding the logarithm first. Multiply the variables directly: abcd = a \cdot a^{(1/2)} \cdot a^{(1/3)} \cdot a^{(1/4)}. When multiplying terms with the same base, add their exponents: abcd = a^{(1 + (1/2) + (1/3) + (1/4))}. Taking \log_a of this expression instantly drops the base a, leaving only the exponent: 1 + (1/2) + (1/3) + (1/4). Common Pitfall: Some students incorrectly look at a = b^2 = c^3 = d^4 and try to substitute coefficients inside the log, such as assuming \log_a b = 2 instead of (1/2). Always be careful when isolating the variable; b^2 = a means taking the square root, hence the fractional power. Final Answer: Therefore, the correct answer is 1 + (1/2) + (1/3) + (1/4).

Question 9

If $\log_{3} x + \log_{9} x^2 + \log_{27} x^3 = 9$, then $x$ equals

Accepted Answer

Concept & Formula: This problem is solved efficiently by normalizing the bases. We use the powerful property that allows us to factor out exponents from both the base and the argument of a logarithm. \log_{a^n} (x^m) = (m/n) \log_a x Step-by-Step Solution: • Given: An equation with three logarithms having different bases (3, 9, 27) but related arguments (x, x^2, x^3). \log_3 x + \log_9 x^2 + \log_{27} x^3 = 9 • Calculation: Express the bases 9 and 27 as powers of 3 to unify the equation. 9 = 3^2 27 = 3^3 \log_3 x + \log_{3^2} x^2 + \log_{3^3} x^3 = 9 • Apply the property \log_{a^n} (x^m) = (m/n) \log_a x to simplify the second and third terms. For the second term: \log_{3^2} x^2 = (2/2) \log_3 x = 1 \log_3 x. For the third term: \log_{3^3} x^3 = (3/3) \log_3 x = 1 \log_3 x. • Substitute these simplified terms back into the original equation. \log_3 x + \log_3 x + \log_3 x = 9 • Add the identical terms together. 3 \log_3 x = 9 • Divide both sides by 3. \log_3 x = 3 • Convert the logarithmic equation to an exponential equation to solve for x. x = 3^3 x = 27 Exam Strategy & Shortcut: Look for symmetry. The terms are \log_{3^1} x^1, \log_{3^2} x^2, and \log_{3^3} x^3. The power of the base perfectly matches the power of the argument in every single term. This means the powers instantly cancel out, leaving you with three identical \log_3 x terms. So, 3 \log_3 x = 9, giving \log_3 x = 3, leading straight to 3^3 = 27. Common Pitfall: A lengthy and error-prone approach is using the standard change-of-base formula ((\log x^2/\log 9)) and converting everything to base 10. While valid, it introduces complex fractions and algebraic manipulation that consumes valuable exam time. Master the exponent factorization property for bases and arguments instead. Final Answer: Therefore, the correct answer is 27.

Question 10

If $\log_{7} \log_{5} (\sqrt{x + 5} + \sqrt{x}) = 0$, what is the value of $x$?

Accepted Answer

Concept & Formula: The core concept is converting a logarithmic equation into an exponential equation. If a logarithm equals a number, you can rewrite it as the base raised to that number equals the argument. Formula: If \log_a(b) = c, then b = a^c. Since this is a nested logarithm, we apply this transformation twice, starting from the outermost base and working our way in. Step-by-Step Solution: Given the equation: \log_{7} \log_{5} (\sqrt{x + 5} + \sqrt{x}) = 0 Step 1: Remove the outermost logarithm (base 7) by converting it to its exponential form. \log_{5} (\sqrt{x + 5} + \sqrt{x}) = 7^0 Since any non-zero number to the power of 0 is 1, this simplifies to: \log_{5} (\sqrt{x + 5} + \sqrt{x}) = 1 Step 2: Remove the next logarithm (base 5) in the exact same way. \sqrt{x + 5} + \sqrt{x} = 5^1 \sqrt{x + 5} + \sqrt{x} = 5 Step 3: Solve for x. The most efficient way is to test the given options to see which one satisfies the simplified equation. Let's test option (c) which is x = 4. \sqrt{4 + 5} + \sqrt{4} \sqrt{9} + \sqrt{4} 3 + 2 = 5 This matches our equation perfectly, confirming that x = 4 is the correct root. Exam Strategy & Shortcut: Option Elimination via Simplification: Nested log problems appear complex but almost always simplify down to a basic algebraic equation rapidly because the right-hand sides are typically 0 or 1. Once reduced to \sqrt{x + 5} + \sqrt{x} = 5, do not square both sides mathematically. Option 4 yields perfect squares (9 and 4), making it the most logical choice to test first and saving precious seconds. Common Pitfall: Students often try to algebraically square the equation \sqrt{x + 5} + \sqrt{x} = 5 immediately. Squaring this expression results in x + 5 + x + 2\sqrt{x^2 + 5x}, which drastically complicates the problem, creates quadratic equations, and eats up valuable exam time. Always check if plugging in the answers provides a quicker shortcut. Final Answer: Therefore, the correct answer is 4.

Question 11

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

Accepted Answer

Concept & Formula: When dealing with common bases, break down the logarithm arguments into their prime factorization. Both 27 and 9 are powers of 3. By finding the fundamental value of \log 3, we can easily compute \log 9. The key rule is the Power Rule of Logarithms. Formula: \log (x^n) = n \log x Step-by-Step Solution: Given the equation: \log 27 = 1.431 Step 1: Rewrite 27 as a power of 3. \log (3^3) = 1.431 Step 2: Apply the power rule to bring the exponent down. 3 \log 3 = 1.431 Step 3: Solve for the base value, \log 3. \log 3 = (1.431/3) \log 3 = 0.477 Step 4: Now find the value of \log 9. Rewrite 9 as 3^2. \log 9 = \log (3^2) \log 9 = 2 \log 3 Step 5: Substitute the value of \log 3 we just found. \log 9 = 2 × 0.477 \log 9 = 0.954 Exam Strategy & Shortcut: Direct Ratio Method: Since both arguments are powers of the same base (3), you can set up a direct ratio of their exponents. \log 27 involves 3^3 (exponent 3) and \log 9 involves 3^2 (exponent 2). Therefore, the value of \log 9 is simply exactly (2/3) of the value of \log 27. \log 9 = (2/3) × 1.431 = 2 × 0.477 = 0.954 Common Pitfall: A frequent mistake is attempting to guess log values or assuming a linear relationship (e.g., thinking \log 9 is one-third of \log 27 just because 9 is one-third of 27). Logarithmic relationships are strictly exponential, so always convert to powers first! Final Answer: Therefore, the correct answer is 0.954.

Question 12

If $\log_{10} 2 = 0.3010$, then $\log_{2} 10$ is equal to

Accepted Answer

Concept & Formula: This problem directly tests your knowledge of the Base-Switching property of logarithms. If you swap the base and the argument of a logarithm, the new logarithmic value is simply the mathematical reciprocal of the original value. Formula: \log_a b = (1/\log_b a) Step-by-Step Solution: Given the equation: \log_{10} 2 = 0.3010 Step 1: Apply the base-switching rule to find \log_{2} 10. \log_{2} 10 = (1/\log_{10) 2} Step 2: Substitute the known numerical value into the denominator. \log_{2} 10 = (1/0.3010) Step 3: Convert the decimal into a fraction to match the provided multiple-choice options. Since 0.3010 can be written as (3010/10000) or (301/1000): \log_{2} 10 = (1/\frac{301){1000}} Step 4: Simplify the complex fraction by flipping the denominator. \log_{2} 10 = (1000/301) Exam Strategy & Shortcut: Reciprocal Decimal Recognition: When you see the base and argument flipped, immediately know you just need the reciprocal: 1 / 0.3010. To calculate quickly without a calculator, ignore the decimal places temporarily. You know you are doing 1 divided by roughly 3/10, so the answer must be around 10/3. Looking at the options, (1000/301) is the exact fractional representation of this reciprocal. Common Pitfall: Students often get confused by the zeros in the decimal and select an incorrectly scaled fraction like (100/301) or mistakenly attempt to use subtraction rules. Keep your decimal places aligned when converting to a fraction: 0.301 = 301/1000. Final Answer: Therefore, the correct answer is (1000/301).

Question 13

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 14

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 15

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 16

If the logarithm of a number is $-3.153$, what are characteristic and mantisa?

Accepted Answer

Concept & Formula: Every logarithm consists of two distinct parts: an integer part called the characteristic, and a fractional/decimal part called the mantissa. The fundamental rule of standard logarithms is that the mantissa must always be a non-negative value (0 \le mantissa < 1). If a logarithm is a negative number, you must restructure it so the decimal part becomes positive. Step-by-Step Solution: Given the logarithm value: -3.153 Step 1: Separate the number into its integer and decimal components. -3.153 = -3 - 0.153 Step 2: Notice that the decimal part (-0.153) is negative, which violates the rule for mantissas. To make it positive, we subtract 1 from the integer part and add 1 to the decimal part. This doesn't change the overall mathematical value of the expression. -3.153 = (-3 - 1) + (1 - 0.153) Step 3: Simplify both parts. The integer part (characteristic) becomes: -3 - 1 = -4 The decimal part (mantissa) becomes: 1 - 0.153 = 0.847 So, the characteristic is -4 (often written as \bar{4}), and the mantissa is 0.847. Exam Strategy & Shortcut: The "Next Lowest Integer" Rule: When dealing with a negative logarithm, the characteristic is always the next integer that is *lower* (more negative) than the given number. For -3.153, the next lowest integer is -4. That is your characteristic instantly. To find the mantissa, just find the complement of the decimal part to 1. (1 - 0.153 = 0.847). Common Pitfall: The most common trap is simply taking the number exactly as it is written, leading students to choose Characteristic = -3 and Mantissa = -0.153. Remember that standard log tables and calculations require a positive mantissa. Final Answer: Therefore, the correct answer is Characteristic = -4, mantissa = 0.847.

Question 17

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

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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 18

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 19

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 20

The number of digits in $4^9 	imes 5^{17}$, when expressed in usual form, is

Accepted Answer

Concept & Strategy: While this could theoretically be solved using logarithms, the most efficient method is basic exponent manipulation. By converting the bases into 2 and 5, we can pair them up to form powers of 10. Every factor of 10 adds exactly one trailing zero to the final number. Step-by-Step Solution: Given the expression: 4^9 × 5^{17} Step 1: Convert the base 4 into base 2 so it can interact with the base 5. (2^2)^9 × 5^{17} Step 2: Multiply the inner and outer exponents for the base 2. 2^{18} × 5^{17} Step 3: To pair the 2s and 5s into 10s, match their exponents. We can split 2^{18} into 2^1 × 2^{17}. 2^1 × 2^{17} × 5^{17} Step 4: Combine the matched powers into a power of 10. 2 × (2 × 5)^{17} 2 × 10^{17} Step 5: Count the digits. The expression 2 × 10^{17} translates to the digit "2" followed by exactly 17 zeros. Total digits = 1 (for the digit 2) + 17 (zeros) = 18 Exam Strategy & Shortcut: Zero Pairing: Recognize instantly that 4^9 is 2^{18}. You have eighteen 2s and seventeen 5s. You can form exactly seventeen 10s (giving you 17 zeros). You are left with one stray 2. A single digit number multiplied by 10^{17} will always have exactly 18 digits. This bypasses logarithmic calculations entirely. Common Pitfall: Attempting to solve this by finding \log(4^9) + \log(5^{17}) manually without a calculator is a massive time sink. While mathematically sound, multiplying \log 4 by 9 and \log 5 by 17 introduces numerous opportunities for arithmetic mistakes compared to basic exponent grouping. Final Answer: Therefore, the correct answer is 18.

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