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Q1: The value of $\log_2 16$ is
A. 4
B. 8
C. 16
D. 2
Correct Answer: 4
Solution: $\text{We know that } \log_a b = x \implies a^x = b.
\text{ Here, } 2^x = 16.
\text{ Since } 16 = 2^4, \text{ we have } x = 4.$
Solution: $\text{We know that } \log_a b = x \implies a^x = b.
\text{ Here, } 2^x = 16.
\text{ Since } 16 = 2^4, \text{ we have } x = 4.$
Q2: The value of $\log_{343} 7$ is
A. $1/3$
B. -3
C. 3
D. 1
Correct Answer: $1/3$
Solution: $\text{We know that } 343 = 7^3.
\text{ So, } \log_{343} 7 = \frac{1}{\log_7 343}.
\text{ Since } \log_7 343 = 3, \text{ we get } \log_{343} 7 = \frac{1}{3}.$
Solution: $\text{We know that } 343 = 7^3.
\text{ So, } \log_{343} 7 = \frac{1}{\log_7 343}.
\text{ Since } \log_7 343 = 3, \text{ we get } \log_{343} 7 = \frac{1}{3}.$
Q3: The value of $\log_5 \left(\frac{125}{625}\right)$ is
A. 5
B. -4
C. 4
D. -5
Correct Answer: -4
Solution: $\text{Simplify } \frac{125}{625} = \frac{1}{5}.
\text{ So, } \log_5 \left(\frac{1}{5}\right) = \log_5 (5^{-1}) = -1.
\text{ Hence, the answer is } -4.$
Solution: $\text{Simplify } \frac{125}{625} = \frac{1}{5}.
\text{ So, } \log_5 \left(\frac{1}{5}\right) = \log_5 (5^{-1}) = -1.
\text{ Hence, the answer is } -4.$
Q4: The value of $\log_{\sqrt{2}} 32$ is
A. 10
B. 5
C. 2
D. 8
Correct Answer: 10
Solution: $\text{Let } x = \log_{\sqrt{2}} 32.
\text{ Then, } (\sqrt{2})^x = 32.
\text{ Since } \sqrt{2} = 2^{1/2}, \text{ we have } (2^{1/2})^x = 2^5.
\text{ Simplifying, } x = 10.$
Solution: $\text{Let } x = \log_{\sqrt{2}} 32.
\text{ Then, } (\sqrt{2})^x = 32.
\text{ Since } \sqrt{2} = 2^{1/2}, \text{ we have } (2^{1/2})^x = 2^5.
\text{ Simplifying, } x = 10.$
Q5: Determine the value of $\log_3 \sqrt{\frac{1}{27}}$
A. 2
B. -2
C. 3
D. -3
Correct Answer: -3
Solution: $\text{Simplify } \sqrt{\frac{1}{27}} = \left(\frac{1}{27}\right)^{1/2} = \left(3^{-3}\right)^{1/2} = 3^{-3/2}.
\text{ So, } \log_3 \sqrt{\frac{1}{27}} = \log_3 (3^{-3/2}) = -\frac{3}{2}.$
Solution: $\text{Simplify } \sqrt{\frac{1}{27}} = \left(\frac{1}{27}\right)^{1/2} = \left(3^{-3}\right)^{1/2} = 3^{-3/2}.
\text{ So, } \log_3 \sqrt{\frac{1}{27}} = \log_3 (3^{-3/2}) = -\frac{3}{2}.$
Q6: The value of $\log_{10} (0.0001)$ is
A. -4
B. -3
C. -2
D. -1
Correct Answer: -4
Solution: $\text{We know that } 0.0001 = 10^{-4}.
\text{ So, } \log_{10} (0.0001) = \log_{10} (10^{-4}) = -4.$
Solution: $\text{We know that } 0.0001 = 10^{-4}.
\text{ So, } \log_{10} (0.0001) = \log_{10} (10^{-4}) = -4.$
Q7: The value of $\log_{0.1} (1000)$ is
A. -3
B. 3
C. -2
D. 2
Correct Answer: -3
Solution: $\text{Let } x = \log_{0.1} (1000).
\text{ Then, } (0.1)^x = 1000.
\text{ Since } 0.1 = 10^{-1}, \text{ we have } (10^{-1})^x = 10^3.
\text{ Simplifying, } x = -3.$
Solution: $\text{Let } x = \log_{0.1} (1000).
\text{ Then, } (0.1)^x = 1000.
\text{ Since } 0.1 = 10^{-1}, \text{ we have } (10^{-1})^x = 10^3.
\text{ Simplifying, } x = -3.$
Q8: What is the value of $[\log_{10} (5 \log_{10} 100)]^2$?
A. 1
B. 5
C. 10
D. 25
Correct Answer: 1
Solution: $\text{First, calculate } \log_{10} 100 = 2.
\text{ Then, } 5 \log_{10} 100 = 5 \cdot 2 = 10.
\text{ Next, } \log_{10} (10) = 1.
\text{ Finally, } [1]^2 = 1.$
Solution: $\text{First, calculate } \log_{10} 100 = 2.
\text{ Then, } 5 \log_{10} 100 = 5 \cdot 2 = 10.
\text{ Next, } \log_{10} (10) = 1.
\text{ Finally, } [1]^2 = 1.$
Q9: The logarithm of 0.0625 to the base 2 is
A. -4
B. -2
C. 0.25
D. 0.5
Correct Answer: -4
Solution: $\text{We know that } 0.0625 = 2^{-4}.
\text{ So, } \log_2 (0.0625) = \log_2 (2^{-4}) = -4.$
Solution: $\text{We know that } 0.0625 = 2^{-4}.
\text{ So, } \log_2 (0.0625) = \log_2 (2^{-4}) = -4.$
Q10: The logarithm of 0.00001 to the base 0.01 is equal to
A. -5
B. 5
C. -1
D. 1
Correct Answer: 5
Solution: $\text{Let } x = \log_{0.01} (0.00001).
\text{ Then, } (0.01)^x = 0.00001.
\text{ Since } 0.01 = 10^{-2}, \text{ we have } (10^{-2})^x = 10^{-5}.
\text{ Simplifying, } x = 5.$
Solution: $\text{Let } x = \log_{0.01} (0.00001).
\text{ Then, } (0.01)^x = 0.00001.
\text{ Since } 0.01 = 10^{-2}, \text{ we have } (10^{-2})^x = 10^{-5}.
\text{ Simplifying, } x = 5.$
Q11: If log₃ x = -2, then x is equal to
A. -9
B. -6
C. 1/9
D. 1/3
Correct Answer: 1/9
Solution: log₃ x = -2 => x = 3⁻² = 1/9.
Solution: log₃ x = -2 => x = 3⁻² = 1/9.
Q12: If log₈ x = 2/3, then the value of x is
A. 4
B. 8
C. 16
D. 2
Correct Answer: 4
Solution: log₈ x = 2/3 => x = 8^(2/3) = (2³)^(2/3) = 2² = 4.
Solution: log₈ x = 2/3 => x = 8^(2/3) = (2³)^(2/3) = 2² = 4.
Q13: If log₈ p = 25 and log₂ q = 5, then
A. p = q¹⁵
B. p² = q³
C. p = q⁵
D. p⁴ = q
Correct Answer: p = q¹⁵
Solution: log₈ p = 25 => p = 8²⁵ = (2³)²⁵ = 2⁷⁵.
log₂ q = 5 => q = 2⁵.
Thus, p = (2⁵)¹⁵ = q¹⁵.
Solution: log₈ p = 25 => p = 8²⁵ = (2³)²⁵ = 2⁷⁵.
log₂ q = 5 => q = 2⁵.
Thus, p = (2⁵)¹⁵ = q¹⁵.
Q14: If logₓ (1/16) = -4, then x is equal to
A. 2
B. 4
C. 8
D. 16
Correct Answer: 2
Solution: logₓ (1/16) = -4 => x⁻⁴ = 1/16 => x⁴ = 16 => x = 2.
Solution: logₓ (1/16) = -4 => x⁻⁴ = 1/16 => x⁴ = 16 => x = 2.
Q15: If logₓ 4 = 0.4, then the value of x is
A. 1
B. 4
C. 16
D. 32
Correct Answer: 32
Solution: logₓ 4 = 0.4 => x⁰·⁴ = 4 => x = 4^(1/0.4) = 4^(5/2) = (2²)^(5/2) = 2⁵ = 32.
Solution: logₓ 4 = 0.4 => x⁰·⁴ = 4 => x = 4^(1/0.4) = 4^(5/2) = (2²)^(5/2) = 2⁵ = 32.
Q16: If log₁₀₀₀₀ x = -1/4, then x is equal to
A. 1/10
B. 1/100
C. 1/1000
D. 1/10000
Correct Answer: 1/10
Solution: log₁₀₀₀₀ x = -1/4 => x = 10000^(-1/4) = (10⁴)^(-1/4) = 10⁻¹ = 1/10.
Solution: log₁₀₀₀₀ x = -1/4 => x = 10000^(-1/4) = (10⁴)^(-1/4) = 10⁻¹ = 1/10.
Q17: If logₓ 4 = 1/2, then x is equal to
A. 16
B. 64
C. 128
D. 256
Correct Answer: 16
Solution: logₓ 4 = 1/2 => x^(1/2) = 4 => x = 4² = 16.
Solution: logₓ 4 = 1/2 => x^(1/2) = 4 => x = 4² = 16.
Q18: If logₓ (0.1) = -1/3, then the value of x is
A. 10
B. 100
C. 1000
D. 1/10
Correct Answer: 1000
Solution: logₓ (0.1) = -1/3 => x^(-1/3) = 0.1 => x = (0.1)^(-3) = 10³ = 1000.
Solution: logₓ (0.1) = -1/3 => x^(-1/3) = 0.1 => x = (0.1)^(-3) = 10³ = 1000.
Q19: If log₃₂ x = 0.8, then x is equal to
A. 16
B. 25.6
C. 10
D. 12.8
Correct Answer: 16
Solution: log₃₂ x = 0.8 => x = 32^(0.8) = (2⁵)^(0.8) = 2⁴ = 16.
Solution: log₃₂ x = 0.8 => x = 32^(0.8) = (2⁵)^(0.8) = 2⁴ = 16.
Q20: If log₂ x = 10 and log y = 100, then y is
A. 2¹⁰
B. 2¹⁰⁰
C. 2¹⁰⁰⁰
D. 2¹⁰⁰⁰⁰
Correct Answer: 2¹⁰⁰
Solution: log₂ x = 10 => x = 2¹⁰.
log y = 100 => y = x¹⁰⁰ = (2¹⁰)¹⁰⁰ = 2¹⁰⁰.
Solution: log₂ x = 10 => x = 2¹⁰.
log y = 100 => y = x¹⁰⁰ = (2¹⁰)¹⁰⁰ = 2¹⁰⁰.
Q21: The value of log(_1/3) 81 is
A. -27
B. -4
C. 4
D. 27
Correct Answer: -4
Solution: Let log(_1/3) 81 = x.
Then, (1/3)^x = 81 => (3^(-1))^x = 3^4 => -x = 4 => x = -4.
Solution: Let log(_1/3) 81 = x.
Then, (1/3)^x = 81 => (3^(-1))^x = 3^4 => -x = 4 => x = -4.
Q22: The value of log₂√₃ (1728) is
A. 3
B. 5
C. 6
D. 9
Correct Answer: 6
Solution: Let log₂√₃ (1728) = x.
Then, (2√3)^x = 1728 => [(2)(3^(1/2))]^x = 1728 => (2^x)(3^(x/2)) = 1728.
Simplifying, x = 6.
Solution: Let log₂√₃ (1728) = x.
Then, (2√3)^x = 1728 => [(2)(3^(1/2))]^x = 1728 => (2^x)(3^(x/2)) = 1728.
Simplifying, x = 6.
Q23: The value of log√8 / log8 is
A. 1/2
B. 2
C. 1/3
D. 3
Correct Answer: 1/2
Solution: log√8 / log8 = log(8^(1/2)) / log8 = (1/2)log8 / log8 = 1/2.
Solution: log√8 / log8 = log(8^(1/2)) / log8 = (1/2)log8 / log8 = 1/2.
Q24: Which of the following statements is not correct?
A. log₁₀ 10 = 1
B. log(2 + 3) = log(2 × 3)
C. log₁₀ 1 = 0
D. log(1 + 2 + 3) = log1 + log2 + log3
Correct Answer: log(2 + 3) = log(2 × 3)
Solution: log(2 + 3) = log5, while log(2 × 3) = log6.
These are not equal.
Hence, this statement is incorrect.
Solution: log(2 + 3) = log5, while log(2 × 3) = log6.
These are not equal.
Hence, this statement is incorrect.
Q25: The value of (6 log₁₀ 1000) / (3 log₁₀ 100) is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 2
Solution: (6 log₁₀ 1000) / (3 log₁₀ 100) = (6 × 3) / (3 × 2) = 18 / 6 = 2.
Solution: (6 log₁₀ 1000) / (3 log₁₀ 100) = (6 × 3) / (3 × 2) = 18 / 6 = 2.
Q26: log₁₀ (10 × 10² × 10³ × ... × 10⁹) is
A. 0
B. 10
C. 20
D. 45
Correct Answer: 45
Solution: log₁₀ (10 × 10² × 10³ × ...
× 10⁹) = log₁₀ (10^(1+2+3+...+9)) = log₁₀ (10^45) = 45.
Solution: log₁₀ (10 × 10² × 10³ × ...
× 10⁹) = log₁₀ (10^(1+2+3+...+9)) = log₁₀ (10^45) = 45.
Q27: The value of log₂ (log₅ 625) is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 2
Solution: log₂ (log₅ 625) = log₂ (log₅ (5⁴)) = log₂ (4 log₅ 5) = log₂ 4 = 2.
Solution: log₂ (log₅ 625) = log₂ (log₅ (5⁴)) = log₂ (4 log₅ 5) = log₂ 4 = 2.
Q28: If log₂ [log₃ (log₂ x)] = 1, then x is
A. 0
B. 12
C. 128
D. 512
Correct Answer: 512
Solution: log₂ [log₃ (log₂ x)] = 1 => log₃ (log₂ x) = 2 => log₂ x = 3² = 9 => x = 2⁹ = 512.
Solution: log₂ [log₃ (log₂ x)] = 1 => log₃ (log₂ x) = 2 => log₂ x = 3² = 9 => x = 2⁹ = 512.
Q29: log₁₀ log₁₀ log₁₀ (10¹⁰¹⁰) is
A. 0
B. 1
C. 10
D. 100
Correct Answer: 1
Solution: log₁₀ log₁₀ log₁₀ (10¹⁰¹⁰) = log₁₀ log₁₀ (10¹⁰ log₁₀ 10) = log₁₀ log₁₀ (10¹⁰) = log₁₀ (10 log₁₀ 10) = log₁₀ 10 = 1.
Solution: log₁₀ log₁₀ log₁₀ (10¹⁰¹⁰) = log₁₀ log₁₀ (10¹⁰ log₁₀ 10) = log₁₀ log₁₀ (10¹⁰) = log₁₀ (10 log₁₀ 10) = log₁₀ 10 = 1.
Q30: The value of log₂ log₂ log₃ (log₃ 27³) is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 0
Solution: log₂ log₂ log₃ (log₃ 27³) = log₂ log₂ log₃ (log₃ (3³)³) = log₂ log₂ log₃ (log₃ 3⁹) = log₂ log₂ log₃ (9 log₃ 3) = log₂ log₂ log₃ 9 = log₂ log₂ (2 log₃ 3) = log₂ log₂ 2 = log₂ 1 = 0.
Solution: log₂ log₂ log₃ (log₃ 27³) = log₂ log₂ log₃ (log₃ (3³)³) = log₂ log₂ log₃ (log₃ 3⁹) = log₂ log₂ log₃ (9 log₃ 3) = log₂ log₂ log₃ 9 = log₂ log₂ (2 log₃ 3) = log₂ log₂ 2 = log₂ 1 = 0.
Q31: The value of log₂ [log₂ {log₄ (log₄ 256⁴)}] is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 0
Solution: log₂ [log₂ {log₄ (log₄ 256⁴)}] = log₂ log₂ [log₄ (log₄ (4⁴)⁴)] = log₂ log₂ [log₄ {log₄ 4¹⁶}] = log₂ log₂ [log₄ (16 log₄ 4)] = log₂ log₂ log₄ 16 = log₂ log₂ (2 log₄ 4) = log₂ log₂ 2 = log₂ 1 = 0.
Solution: log₂ [log₂ {log₄ (log₄ 256⁴)}] = log₂ log₂ [log₄ (log₄ (4⁴)⁴)] = log₂ log₂ [log₄ {log₄ 4¹⁶}] = log₂ log₂ [log₄ (16 log₄ 4)] = log₂ log₂ log₄ 16 = log₂ log₂ (2 log₄ 4) = log₂ log₂ 2 = log₂ 1 = 0.
Q32: If aˣ = bʸ, then log(a/b) is
A. x/y
B. y/x
C. x - y
D. y - x
Correct Answer: y/x
Solution: aˣ = bʸ => log(aˣ) = log(bʸ) => x log a = y log b => log a / log b = y / x => log(a/b) = y/x.
Solution: aˣ = bʸ => log(aˣ) = log(bʸ) => x log a = y log b => log a / log b = y / x => log(a/b) = y/x.
Q33: log 360 is equal to
A. 2 log 2 + 3 log 3
B. 3 log 2 + 2 log 3
C. 3 log 2 + 2 log 3 - log 5
D. 3 log 2 + 2 log 3 + log 5
Correct Answer: 3 log 2 + 2 log 3 + log 5
Solution: log 360 = log (2³ × 3² × 5) = log 2³ + log 3² + log 5 = 3 log 2 + 2 log 3 + log 5.
Solution: log 360 = log (2³ × 3² × 5) = log 2³ + log 3² + log 5 = 3 log 2 + 2 log 3 + log 5.
Q34: log₁₀ (119/13) + log₁₀ (64/26) - log₁₀ (1/10) is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 1
Solution: log₁₀ (119/13) + log₁₀ (64/26) - log₁₀ (1/10) = log₁₀ [(119 × 64) / (13 × 26)] - log₁₀ (1/10) = log₁₀ (10) = 1.
Solution: log₁₀ (119/13) + log₁₀ (64/26) - log₁₀ (1/10) = log₁₀ [(119 × 64) / (13 × 26)] - log₁₀ (1/10) = log₁₀ (10) = 1.
Q35: The value of (1/3 log₁₀ 125 - 2 log₁₀ 4 + log₁₀ 32) is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 1
Solution: (1/3 log₁₀ 125 - 2 log₁₀ 4 + log₁₀ 32) = log₁₀ (125^(1/3)) - log₁₀ (4²) + log₁₀ 32 = log₁₀ 5 - log₁₀ 16 + log₁₀ 32 = log₁₀ (5 × 32 / 16) = log₁₀ 10 = 1.
Solution: (1/3 log₁₀ 125 - 2 log₁₀ 4 + log₁₀ 32) = log₁₀ (125^(1/3)) - log₁₀ (4²) + log₁₀ 32 = log₁₀ 5 - log₁₀ 16 + log₁₀ 32 = log₁₀ (5 × 32 / 16) = log₁₀ 10 = 1.
Q36: log₁₀ (1 + 1/2) + log₁₀ (1 + 1/3) + ... + log₁₀ (1 + 1/198) is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 2
Solution: log₁₀ (1 + 1/2) + log₁₀ (1 + 1/3) + ...
+ log₁₀ (1 + 1/198) = log₁₀ (3/2) + log₁₀ (4/3) + ...
+ log₁₀ (199/198) = log₁₀ (199/2) = log₁₀ 100 = 2.
Solution: log₁₀ (1 + 1/2) + log₁₀ (1 + 1/3) + ...
+ log₁₀ (1 + 1/198) = log₁₀ (3/2) + log₁₀ (4/3) + ...
+ log₁₀ (199/198) = log₁₀ (199/2) = log₁₀ 100 = 2.
Q37: The value of log(9/16) - log(15/32) + log(10/24) is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 0
Solution: log(9/16) - log(15/32) + log(10/24) = log[(9/16) × (32/15) × (10/24)] = log(1) = 0.
Solution: log(9/16) - log(15/32) + log(10/24) = log[(9/16) × (32/15) × (10/24)] = log(1) = 0.
Q38: 2 log₁₀ 5 + log₁₀ 8 - log₁₀ 4 is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 2
Solution: 2 log₁₀ 5 + log₁₀ 8 - log₁₀ 4 = log₁₀ (5²) + log₁₀ 8 - log₁₀ 4 = log₁₀ (25 × 8 / 4) = log₁₀ 50 = 2.
Solution: 2 log₁₀ 5 + log₁₀ 8 - log₁₀ 4 = log₁₀ (5²) + log₁₀ 8 - log₁₀ 4 = log₁₀ (25 × 8 / 4) = log₁₀ 50 = 2.
Q39: log₁₀ 2 + 16 log₁₀ (25/16) + 12 log₁₀ (81/80) + 7 log₁₀ (16/15) is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 1
Solution: log₁₀ 2 + 16 log₁₀ (25/16) + 12 log₁₀ (81/80) + 7 log₁₀ (16/15) = log₁₀ (2 × (25/16)¹⁶ × (81/80)¹² × (16/15)⁷) = log₁₀ 10 = 1.
Solution: log₁₀ 2 + 16 log₁₀ (25/16) + 12 log₁₀ (81/80) + 7 log₁₀ (16/15) = log₁₀ (2 × (25/16)¹⁶ × (81/80)¹² × (16/15)⁷) = log₁₀ 10 = 1.
Q40: If logₐ (ab) = x, then logₐ (b/a) is
A. x/(1-x)
B. x/(x-1)
C. x²
D. x²/(1-x)
Correct Answer: x/(1-x)
Solution: logₐ (ab) = x => logₐ a + logₐ b = x.
Let logₐ b = y.
Then, logₐ a = x - y.
logₐ (b/a) = logₐ b - logₐ a = y - (x - y) = 2y - x.
Solving, we get logₐ (b/a) = x/(1-x).
Solution: logₐ (ab) = x => logₐ a + logₐ b = x.
Let logₐ b = y.
Then, logₐ a = x - y.
logₐ (b/a) = logₐ b - logₐ a = y - (x - y) = 2y - x.
Solving, we get logₐ (b/a) = x/(1-x).
Q41: If logₐ m = x, then log₁/ₐ (1/m) equals
A. -x
B. x
C. 1/x
D. -1/x
Correct Answer: -x
Solution: log₁/ₐ (1/m) = -logₐ m = -x.
Solution: log₁/ₐ (1/m) = -logₐ m = -x.
Q42: If log₁₀ 2 = a and log₁₀ 3 = b, then log₅ 12 equals
A. (a + b)/(1 + a)
B. (a + 2b)/(1 + a)
C. (2a + b)/(1 + a)
D. (2a + b)/(1 - a)
Correct Answer: (2a + b)/(1 - a)
Solution: log₅ 12 = log₁₀ 12 / log₁₀ 5 = (log₁₀ (2² × 3)) / (log₁₀ (10/2)) = (2a + b) / (1 - a).
Solution: log₅ 12 = log₁₀ 12 / log₁₀ 5 = (log₁₀ (2² × 3)) / (log₁₀ (10/2)) = (2a + b) / (1 - a).
Q43: If log 2 = x, log 3 = y, and log 7 = z, then the value of log (4√63) is
A. 2x + (1/2)y + (1/2)z
B. 2x + (1/3)y + (1/3)z
C. 2x + (1/2)y + (1/3)z
D. 2x + (1/3)y + (1/2)z
Correct Answer: 2x + (1/2)y + (1/2)z
Solution: log (4√63) = log 4 + log √63 = log (2²) + log (7 × 3²)^(1/2) = 2x + (1/2)(y + z).
Solution: log (4√63) = log 4 + log √63 = log (2²) + log (7 × 3²)^(1/2) = 2x + (1/2)(y + z).
Q44: If log₄ x + log₂ x = 6, then x is equal to
A. 2
B. 4
C. 8
D. 16
Correct Answer: 16
Solution: log₄ x + log₂ x = 6 => (1/2)log₂ x + log₂ x = 6 => (3/2)log₂ x = 6 => log₂ x = 4 => x = 2⁴ = 16.
Solution: log₄ x + log₂ x = 6 => (1/2)log₂ x + log₂ x = 6 => (3/2)log₂ x = 6 => log₂ x = 4 => x = 2⁴ = 16.
Q45: If log₁₀ (x² - 6x + 10) = 0, then the value of x is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 3
Solution: log₁₀ (x² - 6x + 10) = 0 => x² - 6x + 10 = 10⁰ = 1 => x² - 6x + 9 = 0 => (x - 3)² = 0 => x = 3.
Solution: log₁₀ (x² - 6x + 10) = 0 => x² - 6x + 10 = 10⁰ = 1 => x² - 6x + 9 = 0 => (x - 3)² = 0 => x = 3.
Q46: If log₁₀ x + log₁₀ y = 3 and log₁₀ x - log₁₀ y = 1, then x and y are
A. 10 and 100
B. 100 and 10
C. 1000 and 100
D. 100 and 1000
Correct Answer: 100 and 10
Solution: Adding the equations: 2 log₁₀ x = 4 => log₁₀ x = 2 => x = 10² = 100.
Subtracting: 2 log₁₀ y = 2 => log₁₀ y = 1 => y = 10¹ = 10.
Solution: Adding the equations: 2 log₁₀ x = 4 => log₁₀ x = 2 => x = 10² = 100.
Subtracting: 2 log₁₀ y = 2 => log₁₀ y = 1 => y = 10¹ = 10.
Q47: If log₁₀ x + log₁₀ 5 = 2, then x equals
A. 15
B. 20
C. 25
D. 100
Correct Answer: 20
Solution: log₁₀ x + log₁₀ 5 = 2 => log₁₀ (5x) = 2 => 5x = 10² = 100 => x = 20.
Solution: log₁₀ x + log₁₀ 5 = 2 => log₁₀ (5x) = 2 => 5x = 10² = 100 => x = 20.
Q48: If log₈ x + log₈ (1/x) = 0, then x is
A. 1
B. 2
C. 4
D. 8
Correct Answer: 1
Solution: log₈ x + log₈ (1/x) = log₈ (x × 1/x) = log₈ 1 = 0.
Thus, x can be any positive number, but the simplest solution is x = 1.
Solution: log₈ x + log₈ (1/x) = log₈ (x × 1/x) = log₈ 1 = 0.
Thus, x can be any positive number, but the simplest solution is x = 1.
Q49: If log₁₀ 125 + log₁₀ 8 = x, then x is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 3
Solution: log₁₀ 125 + log₁₀ 8 = log₁₀ (125 × 8) = log₁₀ 1000 = log₁₀ (10³) = 3.
Solution: log₁₀ 125 + log₁₀ 8 = log₁₀ (125 × 8) = log₁₀ 1000 = log₁₀ (10³) = 3.
Q50: The value of (log₉ 27 + log₈ 32) is
A. 19/6
B. 7/2
C. 13/6
D. 5/2
Correct Answer: 19/6
Solution: log₉ 27 = log₃² (3³) = 3/2.
log₈ 32 = log₂³ (2⁵) = 5/3.
Adding: 3/2 + 5/3 = 19/6.
Solution: log₉ 27 = log₃² (3³) = 3/2.
log₈ 32 = log₂³ (2⁵) = 5/3.
Adding: 3/2 + 5/3 = 19/6.
Q51: (log₅ 3) × (log₃ 625) equals
A. 2
B. 3
C. 4
D. 5
Correct Answer: 4
Solution: (log₅ 3) × (log₃ 625) = log₅ 625 = log₅ (5⁴) = 4.
Solution: (log₅ 3) × (log₃ 625) = log₅ 625 = log₅ (5⁴) = 4.
Q52: (log₅ 5) × (log₄ 9) × (log₃ 2) is equal to
A. 1
B. 2
C. 3
D. 4
Correct Answer: 1
Solution: (log₅ 5) × (log₄ 9) × (log₃ 2) = 1 × (log₂² 3²) × (log₃ 2) = 1 × 1 × 1 = 1.
Solution: (log₅ 5) × (log₄ 9) × (log₃ 2) = 1 × (log₂² 3²) × (log₃ 2) = 1 × 1 × 1 = 1.
Q53: If log₁₂ 27 = a, then log₆ 16 is
A. (3-a)/(4(3+a))
B. (3+a)/(4(3-a))
C. (4(3-a))/(3+a)
D. (4(3+a))/(3-a)
Correct Answer: (4(3-a))/(3+a)
Solution: log₁₂ 27 = a => log₃ 27 / log₃ 12 = a => 3 / (2 + log₃ 2) = a.
Solving gives log₆ 16 = (4(3-a))/(3+a).
Solution: log₁₂ 27 = a => log₃ 27 / log₃ 12 = a => 3 / (2 + log₃ 2) = a.
Solving gives log₆ 16 = (4(3-a))/(3+a).
Q54: If log₁₀ 5 + log₁₀ (5x + 1) = log₁₀ (x + 5) + 1, then x is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 3
Solution: log₁₀ 5 + log₁₀ (5x + 1) = log₁₀ (x + 5) + log₁₀ 10 => log₁₀ [5(5x + 1)] = log₁₀ [10(x + 5)].
Simplifying: 5(5x + 1) = 10(x + 5) => x = 3.
Solution: log₁₀ 5 + log₁₀ (5x + 1) = log₁₀ (x + 5) + log₁₀ 10 => log₁₀ [5(5x + 1)] = log₁₀ [10(x + 5)].
Simplifying: 5(5x + 1) = 10(x + 5) => x = 3.
Q55: If log₅ (x² + x) - log₅ (x + 1) = 2, then x is
A. 1
B. 5
C. 25
D. 125
Correct Answer: 25
Solution: log₅ [(x² + x)/(x + 1)] = 2 => (x² + x)/(x + 1) = 5² = 25.
Solving: x = 25.
Solution: log₅ [(x² + x)/(x + 1)] = 2 => (x² + x)/(x + 1) = 5² = 25.
Solving: x = 25.
Q56: ~ (log x + log y) will equal log (x/y) if
A. x = y
B. xy = 1
C. x = 1/y
D. y = 1/x
Correct Answer: x = y
Solution: ~ (log x + log y) = log (x/y) => log (xy) = log (x/y).
This implies x = y.
Solution: ~ (log x + log y) = log (x/y) => log (xy) = log (x/y).
This implies x = y.
Q57: The value of [log₆₀ 3 + log₆₀ 4 + log₆₀ 5] is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 1
Solution: log₆₀ 3 + log₆₀ 4 + log₆₀ 5 = log₆₀ (3 × 4 × 5) = log₆₀ 60 = 1.
Solution: log₆₀ 3 + log₆₀ 4 + log₆₀ 5 = log₆₀ (3 × 4 × 5) = log₆₀ 60 = 1.
Q58: The value of (log₃ 4) × (log₄ 5) × (log₅ 6) × (log₆ 7) × (log₇ 8) × (log₈ 9) is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 2
Solution: Using the change of base formula: (log₃ 4) × (log₄ 5) × ...
× (log₈ 9) = log₃ 9 = 2.
Solution: Using the change of base formula: (log₃ 4) × (log₄ 5) × ...
× (log₈ 9) = log₃ 9 = 2.
Q59: The value of 16^(log₄ 5) is
A. 25
B. 50
C. 75
D. 100
Correct Answer: 25
Solution: 16^(log₄ 5) = (4²)^(log₄ 5) = 4^(2 × log₄ 5) = (4^(log₄ 5))² = 5² = 25.
Solution: 16^(log₄ 5) = (4²)^(log₄ 5) = 4^(2 × log₄ 5) = (4^(log₄ 5))² = 5² = 25.
Q60: If log x + log y = log (x + y), then
A. x = y
B. x = 1/y
C. x = 0
D. y = 0
Correct Answer: x = y
Solution: log x + log y = log (x + y) => log (xy) = log (x + y).
This implies x = y.
Solution: log x + log y = log (x + y) => log (xy) = log (x + y).
This implies x = y.
Q61: If log (a/b) + log (b/a) = log (a + b), then
A. a = b
B. a = 1/b
C. a = 0
D. b = 0
Correct Answer: a = b
Solution: log (a/b) + log (b/a) = log (a + b) => log 1 = log (a + b).
This implies a = b.
Solution: log (a/b) + log (b/a) = log (a + b) => log 1 = log (a + b).
This implies a = b.
Q62: The value of [1/(logₐ bc) + 1/(logₐ ca) + 1/(logₐ ab)] is
A. 0
B. 1
C. 2
D. 3
Correct Answer: 1
Solution: Using the property of logarithms: [1/(logₐ bc) + 1/(logₐ ca) + 1/(logₐ ab)] = logₐ a + logₐ b + logₐ c = logₐ (abc) = 1.
Solution: Using the property of logarithms: [1/(logₐ bc) + 1/(logₐ ca) + 1/(logₐ ab)] = logₐ a + logₐ b + logₐ c = logₐ (abc) = 1.
Q63: If log x - 5 log 3 = -2, then x equals
A. 0.43
B. 2.43
C. 4.43
D. 6.43
Correct Answer: 2.43
Solution: log x - 5 log 3 = -2 => log x = log 3⁵ - 2 => x = 3⁵ / 10² = 243 / 100 = 2.43.
Solution: log x - 5 log 3 = -2 => log x = log 3⁵ - 2 => x = 3⁵ / 10² = 243 / 100 = 2.43.
Q64: If a = b² = c³ = d⁴, then logₐ (abcd) equals
A. 1 + 1/2 + 1/3 + 1/4
B. 1 + 2 + 3 + 4
C. 1/2 + 1/3 + 1/4
D. 2 + 3 + 4
Correct Answer: 1 + 1/2 + 1/3 + 1/4
Solution: Let a = k, b = k^(1/2), c = k^(1/3), d = k^(1/4).
Then, logₐ (abcd) = logₐ (k × k^(1/2) × k^(1/3) × k^(1/4)) = 1 + 1/2 + 1/3 + 1/4.
Solution: Let a = k, b = k^(1/2), c = k^(1/3), d = k^(1/4).
Then, logₐ (abcd) = logₐ (k × k^(1/2) × k^(1/3) × k^(1/4)) = 1 + 1/2 + 1/3 + 1/4.
Q65: If log₃ x + log₉ x² + log₂₇ x³ = 9, then x is
A. 9
B. 18
C. 27
D. 36
Correct Answer: 27
Solution: log₃ x + log₉ x² + log₂₇ x³ = 9 => log₃ x + (2/2)log₃ x + (3/3)log₃ x = 9 => 3 log₃ x = 9 => log₃ x = 3 => x = 3³ = 27.
Solution: log₃ x + log₉ x² + log₂₇ x³ = 9 => log₃ x + (2/2)log₃ x + (3/3)log₃ x = 9 => 3 log₃ x = 9 => log₃ x = 3 => x = 3³ = 27.
Q66: If log₇ log₅ (√x + 5 + √x) = 0, then x is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 4
Solution: log₇ log₅ (√x + 5 + √x) = 0 => log₅ (√x + 5 + √x) = 1 => √x + 5 + √x = 5.
Solving: x = 4.
Solution: log₇ log₅ (√x + 5 + √x) = 0 => log₅ (√x + 5 + √x) = 1 => √x + 5 + √x = 5.
Solving: x = 4.
Q67: If a = log₈ 225 and b = log₂ 15, then a in terms of b is
A. 2b/3
B. 3b/2
C. b/2
D. 2b
Correct Answer: 2b/3
Solution: a = log₈ 225 = log₂³ (15²) = (2/3)log₂ 15 = (2/3)b.
Solution: a = log₈ 225 = log₂³ (15²) = (2/3)log₂ 15 = (2/3)b.
Q68: If log x - 5 log 3 = -2, then x equals
A. 0.43
B. 2.43
C. 4.43
D. 6.43
Correct Answer: 2.43
Solution: log x - 5 log 3 = -2 => log x = log 3⁵ - 2 => x = 3⁵ / 10² = 243 / 100 = 2.43.
Solution: log x - 5 log 3 = -2 => log x = log 3⁵ - 2 => x = 3⁵ / 10² = 243 / 100 = 2.43.
Q69: If a = b² = c³ = d⁴, then logₐ (abcd) equals
A. 1 + 1/2 + 1/3 + 1/4
B. 1 + 2 + 3 + 4
C. 1/2 + 1/3 + 1/4
D. 2 + 3 + 4
Correct Answer: 1 + 1/2 + 1/3 + 1/4
Solution: Let a = k, b = k^(1/2), c = k^(1/3), d = k^(1/4).
Then, logₐ (abcd) = logₐ (k × k^(1/2) × k^(1/3) × k^(1/4)) = 1 + 1/2 + 1/3 + 1/4.
Solution: Let a = k, b = k^(1/2), c = k^(1/3), d = k^(1/4).
Then, logₐ (abcd) = logₐ (k × k^(1/2) × k^(1/3) × k^(1/4)) = 1 + 1/2 + 1/3 + 1/4.
Q70: If log₃ x + log₉ x² + log₂₇ x³ = 9, then x is
A. 9
B. 18
C. 27
D. 36
Correct Answer: 27
Solution: log₃ x + log₉ x² + log₂₇ x³ = 9 => log₃ x + (2/2)log₃ x + (3/3)log₃ x = 9 => 3 log₃ x = 9 => log₃ x = 3 => x = 3³ = 27.
Solution: log₃ x + log₉ x² + log₂₇ x³ = 9 => log₃ x + (2/2)log₃ x + (3/3)log₃ x = 9 => 3 log₃ x = 9 => log₃ x = 3 => x = 3³ = 27.
Q71: If log₇ log₅ (√x + 5 + √x) = 0, then x is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 4
Solution: log₇ log₅ (√x + 5 + √x) = 0 => log₅ (√x + 5 + √x) = 1 => √x + 5 + √x = 5.
Solving: x = 4.
Solution: log₇ log₅ (√x + 5 + √x) = 0 => log₅ (√x + 5 + √x) = 1 => √x + 5 + √x = 5.
Solving: x = 4.
Q72: If a = log₈ 225 and b = log₂ 15, then a in terms of b is
A. 2b/3
B. 3b/2
C. b/2
D. 2b
Correct Answer: 2b/3
Solution: a = log₈ 225 = log₂³ (15²) = (2/3)log₂ 15 = (2/3)b.
Solution: a = log₈ 225 = log₂³ (15²) = (2/3)log₂ 15 = (2/3)b.
Q73: If log 27 = 1.431, then the value of log 9 is
A. 0.934
B. 0.945
C. 0.954
D. 0.958
Correct Answer: 0.954
Solution: log 27 = 1.431 => log (3³) = 1.431 => 3 log 3 = 1.431 => log 3 = 0.477.
log 9 = log (3²) = 2 log 3 = 2 × 0.477 = 0.954.
Solution: log 27 = 1.431 => log (3³) = 1.431 => 3 log 3 = 1.431 => log 3 = 0.477.
log 9 = log (3²) = 2 log 3 = 2 × 0.477 = 0.954.
Q74: If log₁₀ 2 = 0.3010, then log₂ 10 is
A. 3.322
B. 3.333
C. 3.344
D. 3.355
Correct Answer: 3.322
Solution: log₂ 10 = 1 / log₁₀ 2 = 1 / 0.3010 ≈ 3.322.
Solution: log₂ 10 = 1 / log₁₀ 2 = 1 / 0.3010 ≈ 3.322.
Q75: If log₁₀ 2 = 0.3010, then log₁₀ 5 is
A. 0.699
B. 0.7
C. 0.701
D. 0.702
Correct Answer: 0.699
Solution: log₁₀ 5 = log₁₀ (10/2) = log₁₀ 10 - log₁₀ 2 = 1 - 0.3010 = 0.699.
Solution: log₁₀ 5 = log₁₀ (10/2) = log₁₀ 10 - log₁₀ 2 = 1 - 0.3010 = 0.699.
Q76: If log₁₀ 2 = 0.3010, then log₁₀ 80 is
A. 1.602
B. 1.903
C. 3.903
D. None of these
Correct Answer: 1.903
Solution: log₁₀ 80 = log₁₀ (8 × 10) = log₁₀ 8 + log₁₀ 10 = 3 log₁₀ 2 + 1 = 3 × 0.3010 + 1 = 1.9030.
Solution: log₁₀ 80 = log₁₀ (8 × 10) = log₁₀ 8 + log₁₀ 10 = 3 log₁₀ 2 + 1 = 3 × 0.3010 + 1 = 1.9030.
Q77: If log 3 = 0.477 and (1000)^x = 3, then x equals
A. 0.0159
B. 0.0477
C. 0.159
D. 10
Correct Answer: 0.159
Solution: (1000)^x = 3 => x log 1000 = log 3 => 3x = 0.477 => x = 0.477 / 3 = 0.159.
Solution: (1000)^x = 3 => x log 1000 = log 3 => 3x = 0.477 => x = 0.477 / 3 = 0.159.
Q78: If log₁₀ 2 = 0.3010, then log₁₀ 25 is
A. 0.602
B. 1.204
C. 1.398
D. 1.505
Correct Answer: 1.398
Solution: log₁₀ 25 = log₁₀ (100/4) = log₁₀ 100 - log₁₀ 4 = 2 - 2 log₁₀ 2 = 2 - 2 × 0.3010 = 1.3980.
Solution: log₁₀ 25 = log₁₀ (100/4) = log₁₀ 100 - log₁₀ 4 = 2 - 2 log₁₀ 2 = 2 - 2 × 0.3010 = 1.3980.
Q79: If log₁₀ 20 = 1.3010 and log₁₀ 30 = 1.4771, then log₁₀ (60000) is
A. 0.7781
B. 1.7781
C. 2.7781
D. 4.7781
Correct Answer: 4.7781
Solution: log₁₀ (60000) = log₁₀ (20 × 30 × 100) = log₁₀ 20 + log₁₀ 30 + log₁₀ 100 = 1.3010 + 1.4771 + 2 = 4.7781.
Solution: log₁₀ (60000) = log₁₀ (20 × 30 × 100) = log₁₀ 20 + log₁₀ 30 + log₁₀ 100 = 1.3010 + 1.4771 + 2 = 4.7781.
Q80: If log 2 = 0.3010 and log 3 = 0.4771, then log₅ 512 is
A. 2.87
B. 3.876
C. 4.876
D. 5.876
Correct Answer: 3.876
Solution: log₅ 512 = log 512 / log 5 = log (2⁹) / log (10/2) = 9 log 2 / (1 - log 2) = 9 × 0.3010 / (1 - 0.3010) ≈ 3.876.
Solution: log₅ 512 = log 512 / log 5 = log (2⁹) / log (10/2) = 9 log 2 / (1 - log 2) = 9 × 0.3010 / (1 - 0.3010) ≈ 3.876.
Q81: If log₁₀ 3 = 0.4771 and log₁₀ 7 = 0.8451, then log₁₀ (23/3) is
A. 0.368
B. 1.368
C. 2.368
D. 3.368
Correct Answer: 1.368
Solution: log₁₀ (23/3) = log₁₀ 7 + log₁₀ 10 - log₁₀ 3 = 0.8451 + 1 - 0.4771 = 1.368.
Solution: log₁₀ (23/3) = log₁₀ 7 + log₁₀ 10 - log₁₀ 3 = 0.8451 + 1 - 0.4771 = 1.368.
Q82: If log₁₀ 2 = 0.3010 and log₁₀ 3 = 0.4771, then log₁₀ 1.5 is
A. 0.1761
B. 0.3561
C. 0.7161
D. 0.9161
Correct Answer: 0.1761
Solution: log₁₀ 1.5 = log₁₀ (3/2) = log₁₀ 3 - log₁₀ 2 = 0.4771 - 0.3010 = 0.1761.
Solution: log₁₀ 1.5 = log₁₀ (3/2) = log₁₀ 3 - log₁₀ 2 = 0.4771 - 0.3010 = 0.1761.
Q83: If log₁₀ 2 = 0.3010 and log₁₀ 7 = 0.8451, then log₁₀ 2.8 is
A. 0.4471
B. 1.4471
C. 2.4471
D. None of these
Correct Answer: 0.4471
Solution: log₁₀ 2.8 = log₁₀ (28/10) = log₁₀ 28 - log₁₀ 10 = log₁₀ (7 × 4) - 1 = log₁₀ 7 + 2 log₁₀ 2 - 1 = 0.8451 + 2 × 0.3010 - 1 = 0.4471.
Solution: log₁₀ 2.8 = log₁₀ (28/10) = log₁₀ 28 - log₁₀ 10 = log₁₀ (7 × 4) - 1 = log₁₀ 7 + 2 log₁₀ 2 - 1 = 0.8451 + 2 × 0.3010 - 1 = 0.4471.
Q84: If log (0.57) = 1.756, then log 57 + log (0.57)³ + log √0.57 is
A. 0.902
B. 1.902
C. 2.902
D. 3.902
Correct Answer: 0.902
Solution: log 57 + log (0.57)³ + log √0.57 = log 57 + 3 log 0.57 + (1/2)log 0.57 = 1.756 + 3 × 1.756 + (1/2) × 1.756 = 0.902.
Solution: log 57 + log (0.57)³ + log √0.57 = log 57 + 3 log 0.57 + (1/2)log 0.57 = 1.756 + 3 × 1.756 + (1/2) × 1.756 = 0.902.
Q85: If the logarithm of a number is -3.153, what are characteristic and mantissa?
A. Char = -4, Mant = 0.847
B. Char = -3, Mant = -0.153
C. Char = 4, Mant = -0.847
D. Char = 3, Mant = -0.153
Correct Answer: Char = -4, Mant = 0.847
Solution: log x = -3.153 = -4 + 0.847.
Characteristic = -4, Mantissa = 0.847.
Solution: log x = -3.153 = -4 + 0.847.
Characteristic = -4, Mantissa = 0.847.
Q86: If log 2 = 0.30103, the number of digits in 2⁶⁴ is
A. 18
B. 19
C. 20
D. 21
Correct Answer: 20
Solution: log 2⁶⁴ = 64 × log 2 = 64 × 0.30103 = 19.26592.
Characteristic = 19.
Number of digits = 19 + 1 = 20.
Solution: log 2⁶⁴ = 64 × log 2 = 64 × 0.30103 = 19.26592.
Characteristic = 19.
Number of digits = 19 + 1 = 20.
Q87: If log 2 = 0.30103, the number of digits in 4⁵⁰ is
A. 30
B. 31
C. 100
D. 200
Correct Answer: 31
Solution: log 4⁵⁰ = 50 × log 4 = 50 × 2 × log 2 = 100 × 0.30103 = 30.103.
Characteristic = 30.
Number of digits = 30 + 1 = 31.
Solution: log 4⁵⁰ = 50 × log 4 = 50 × 2 × log 2 = 100 × 0.30103 = 30.103.
Characteristic = 30.
Number of digits = 30 + 1 = 31.
Q88: If log 2 = 0.30103, the number of digits in 5²⁰ is
A. 14
B. 16
C. 18
D. 25
Correct Answer: 14
Solution: log 5²⁰ = 20 × log 5 = 20 × (1 - log 2) = 20 × (1 - 0.30103) = 13.98.
Characteristic = 13.
Number of digits = 13 + 1 = 14.
Solution: log 5²⁰ = 20 × log 5 = 20 × (1 - log 2) = 20 × (1 - 0.30103) = 13.98.
Characteristic = 13.
Number of digits = 13 + 1 = 14.
Q89: If log 2 = 0.30103, log 3 = 0.47712, then the number of digits in 6²⁰ is
A. 15
B. 16
C. 17
D. 18
Correct Answer: 16
Solution: log 6²⁰ = 20 × log 6 = 20 × (log 2 + log 3) = 20 × (0.30103 + 0.47712) = 15.563.
Characteristic = 15.
Number of digits = 15 + 1 = 16.
Solution: log 6²⁰ = 20 × log 6 = 20 × (log 2 + log 3) = 20 × (0.30103 + 0.47712) = 15.563.
Characteristic = 15.
Number of digits = 15 + 1 = 16.
Q90: The number of digits in 4⁹ × 5¹⁷ is
A. 16
B. 17
C. 18
D. 19
Correct Answer: 18
Solution: log (4⁹ × 5¹⁷) = 9 log 4 + 17 log 5 = 18 log 2 + 17 log 5 = 18 × 0.3010 + 17 × 0.6990 = 17.3010.
Characteristic = 17.
Number of digits = 17 + 1 = 18.
Solution: log (4⁹ × 5¹⁷) = 9 log 4 + 17 log 5 = 18 log 2 + 17 log 5 = 18 × 0.3010 + 17 × 0.6990 = 17.3010.
Characteristic = 17.
Number of digits = 17 + 1 = 18.
Q91: If log₃ (log₅ (x + 5 + √x)) = 0, then x is
A. 1
B. 2
C. 3
D. 4
Correct Answer: 4
Solution: log₃ (log₅ (x + 5 + √x)) = 0 => log₅ (x + 5 + √x) = 1 => x + 5 + √x = 5.
Solving: x = 4.
Solution: log₃ (log₅ (x + 5 + √x)) = 0 => log₅ (x + 5 + √x) = 1 => x + 5 + √x = 5.
Solving: x = 4.
Q92: If log₁₀ a = p and log₁₀ b = q, then log₁₀ (a²b²) is
A. p² + q²
B. p² - q²
C. p²q²
D. p²/2
Correct Answer: p² + q²
Solution: log₁₀ (a²b²) = log₁₀ a² + log₁₀ b² = 2 log₁₀ a + 2 log₁₀ b = 2p + 2q = p² + q².
Solution: log₁₀ (a²b²) = log₁₀ a² + log₁₀ b² = 2 log₁₀ a + 2 log₁₀ b = 2p + 2q = p² + q².
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