{
  "title": "Laws of Logarithms - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Which law says logₐ(mn) = logₐ m + logₐ n? 📚", "options": {"A": "Power Law", "B": "Quotient Law", "C": "Product Law", "D": "Change of Base"}, "correct_answer": "C"},
    {"id": 2, "difficulty": "easy", "question": "Simplify: log 3 + log 4 using laws. ➕", "options": {"A": "log 7", "B": "log 12", "C": "log 1", "D": "log 34"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "What is logₐ (m/n) equal to? ➗", "options": {"A": "logₐ m × logₐ n", "B": "logₐ m - logₐ n", "C": "logₐ m + logₐ n", "D": "logₐ n - logₐ m"}, "correct_answer": "B"},
    {"id": 4, "difficulty": "easy", "question": "Simplify: log 10 - log 2. ✂️", "options": {"A": "log 8", "B": "log 5", "C": "log 20", "D": "log 12"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "According to power law, log mⁿ = ? ⚡", "options": {"A": "n log m", "B": "m log n", "C": "log m × log n", "D": "log (m+n)"}, "correct_answer": "A"},
    {"id": 6, "difficulty": "easy", "question": "Simplify: 2 log 5. 🔢", "options": {"A": "log 10", "B": "log 25", "C": "log 7", "D": "log 5²"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "Combine: log 6 - log 3. 🌀", "options": {"A": "log 2", "B": "log 3", "C": "log 9", "D": "log 18"}, "correct_answer": "A"},
    {"id": 8, "difficulty": "easy", "question": "What is logₐ 1 always equal to? 🎯", "options": {"A": "a", "B": "1", "C": "0", "D": "10"}, "correct_answer": "C"},
    {"id": 9, "difficulty": "easy", "question": "Use power law: log √x = ? 🌱", "options": {"A": "(1/2) log x", "B": "2 log x", "C": "log x²", "D": "√(log x)"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Simplify: log 2 + log 3 + log 5. 📊", "options": {"A": "log 10", "B": "log 30", "C": "log 60", "D": "log 25"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "Expand: log (ab) using laws. 📖", "options": {"A": "log a × log b", "B": "log a + log b", "C": "log a - log b", "D": "log b - log a"}, "correct_answer": "B"},
    {"id": 12, "difficulty": "easy", "question": "If log 5 = k, what is log 25? 🔍", "options": {"A": "k²", "B": "2k", "C": "k+2", "D": "5k"}, "correct_answer": "B"},
    {"id": 13, "difficulty": "easy", "question": "Combine into one log: log p - log q. 🧩", "options": {"A": "log (p/q)", "B": "log (pq)", "C": "log (q/p)", "D": "log (p-q)"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Expand: log (x/y). 🌿", "options": {"A": "log x × log y", "B": "log x + log y", "C": "log x - log y", "D": "log y - log x"}, "correct_answer": "C"},
    {"id": 15, "difficulty": "easy", "question": "Simplify: 3 log 2. 🔨", "options": {"A": "log 6", "B": "log 8", "C": "log 5", "D": "log 2³"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "If log a + log b = log 12, then ab = ? 🤔", "options": {"A": "log 12", "B": "12", "C": "a+b", "D": "144"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "Use power law: log x³ = ? 📦", "options": {"A": "3 log x", "B": "x log 3", "C": "log 3x", "D": "(log x)³"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Simplify: log 100 - log 10. ⚖️", "options": {"A": "log 90", "B": "1", "C": "log 10", "D": "10"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "Combine: 2 log a + 3 log b. ✨", "options": {"A": "log (2a+3b)", "B": "log (a²b³)", "C": "log (6ab)", "D": "log (a+b)⁵"}, "correct_answer": "B"},
    {"id": 20, "difficulty": "easy", "question": "What is logₐ a equal to? ✅", "options": {"A": "0", "B": "1", "C": "a", "D": "10"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "Express as a single log: 2 log x + 0.5 log y - 3 log z. 🧮", "options": {"A": "log (x²√y / z³)", "B": "log (2x + 0.5y - 3z)", "C": "log (x²y⁰·⁵z³)", "D": "log (x²√y z³)"}, "correct_answer": "A"},
    {"id": 22, "difficulty": "medium", "question": "If log 2 = a, log 3 = b, find log 72 in terms of a and b. 🔢", "options": {"A": "2a + 3b", "B": "3a + 2b", "C": "a² + b³", "D": "6ab"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "Solve: log (x+5) - log (x-1) = log 3. ⚖️", "options": {"A": "2", "B": "4", "C": "6", "D": "8"}, "correct_answer": "B"},
    {"id": 24, "difficulty": "medium", "question": "Simplify: log (x² - 4) - log (x+2). ✂️", "options": {"A": "log (x-2)", "B": "log (x+2)", "C": "log (x²-2)", "D": "log 2"}, "correct_answer": "A"},
    {"id": 25, "difficulty": "medium", "question": "Given log 5 = 0.6990, find log √125. 🌳", "options": {"A": "1.0485", "B": "0.3495", "C": "1.3980", "D": "0.6990"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "Combine: 1/2 log 16 + 1/3 log 27. 📐", "options": {"A": "log 6", "B": "log 12", "C": "log 24", "D": "log 36"}, "correct_answer": "B"},
    {"id": 27, "difficulty": "medium", "question": "If log (a+b) = log a + log b, find a relation between a and b. 🤝", "options": {"A": "a = b", "B": "a + b = ab", "C": "a = 1/b", "D": "a - b = ab"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Expand using laws: log (p²q³/r). 📖", "options": {"A": "2 log p + 3 log q - log r", "B": "log p² + log q³ - log r", "C": "log (2p+3q-r)", "D": "Both A & B"}, "correct_answer": "D"},
    {"id": 29, "difficulty": "medium", "question": "Simplify: log (100x) - log (10x). 🌀", "options": {"A": "log 10", "B": "1", "C": "log (90x)", "D": "log 10x"}, "correct_answer": "B"},
    {"id": 30, "difficulty": "medium", "question": "Given log 2 = 0.3010, find log 50. 📍", "options": {"A": "1.6990", "B": "2.6990", "C": "0.6990", "D": "1.3010"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "Solve: 2 log x = log 36. ⚡", "options": {"A": "6", "B": "18", "C": "36", "D": "±6"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If log (x/y) + log (y/x) = 0, prove it. What is its value? ✅", "options": {"A": "0", "B": "1", "C": "log 1", "D": "Both A & C"}, "correct_answer": "D"},
    {"id": 33, "difficulty": "medium", "question": "Express log (√a × ∛b) as sum of logs. 🌿", "options": {"A": "(1/2)log a + (1/3)log b", "B": "log a/2 + log b/3", "C": "log √a + log ∛b", "D": "All of these"}, "correct_answer": "D"},
    {"id": 34, "difficulty": "medium", "question": "Simplify: (log 8)/(log 2) using change of base. 🔄", "options": {"A": "3", "B": "4", "C": "log 4", "D": "8"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "If log x + log (x-3) = 1, find x. (Assume base 10) 🧩", "options": {"A": "5", "B": "2", "C": "5 or -2", "D": "5 only"}, "correct_answer": "D"},
    {"id": 36, "difficulty": "medium", "question": "Prove: log (1+2+3) = log 1 + log 2 + log 3. True or False? ❓", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot say"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "Given log 3 = 0.4771, find log 0.027. 📉", "options": {"A": "-2.5229", "B": "-1.5229", "C": "1.4771", "D": "0.0337"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "Combine: 3 log a - 2 log b + log c. 🧱", "options": {"A": "log (3a - 2b + c)", "B": "log (a³c / b²)", "C": "log (a³/b²) + log c", "D": "Both B & C"}, "correct_answer": "D"},
    {"id": 39, "difficulty": "medium", "question": "Solve: log₂ (x+1) + log₂ (x-1) = 3. ⚔️", "options": {"A": "3", "B": "2", "C": "±3", "D": "3 only"}, "correct_answer": "D"},
    {"id": 40, "difficulty": "medium", "question": "Simplify: log (a/b) + log (b/c) + log (c/a). 🔁", "options": {"A": "log 1", "B": "0", "C": "log (abc)", "D": "Both A & B"}, "correct_answer": "D"},
    {"id": 41, "difficulty": "hard", "question": "Prove: (log a / log b) × (log b / log c) × (log c / log a) = 1. 🏆", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Only if a=b=c"}, "correct_answer": "A"},
    {"id": 42, "difficulty": "hard", "question": "If a² + b² = 7ab, prove that log ((a+b)/3) = (1/2)(log a + log b). Find its value if a=4, b=1? ✅", "options": {"A": "0", "B": "log 5/3", "C": "1/2 log 4", "D": "log 2"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "hard", "question": "Solve: log₄ (x) + log₈ (x) = 4. 🔎", "options": {"A": "16", "B": "32", "C": "64", "D": "256"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "hard", "question": "If x = log 2, y = log 3, express log (√1.2) in terms of x and y. 🌌", "options": {"A": "(x + y - 1)/2", "B": "(1 - x - y)/2", "C": "(2x + y - 1)/2", "D": "(x + 2y - 1)/2"}, "correct_answer": "A"},
    {"id": 45, "difficulty": "hard", "question": "Simplify: log (tan1°) + log (tan2°) + ... + log (tan89°). 🤯", "options": {"A": "0", "B": "1", "C": "log 1", "D": "Both A & C"}, "correct_answer": "D"},
    {"id": 46, "difficulty": "hard", "question": "Given log (x²y³) = 1 and log (x/y) = 2, find log x and log y. 📊", "options": {"A": "log x=1, log y=-1", "B": "log x=2, log y=1", "C": "log x=1.6, log y=-0.4", "D": "No solution"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "hard", "question": "If a, b, c are in G.P., prove that log a, log b, log c are in A.P. ✅", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Only if a=b=c"}, "correct_answer": "A"},
    {"id": 48, "difficulty": "hard", "question": "Solve for x: log (3x+2) - log (3x-2) = log 5. 🌀", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "Prove: (1/log₂ 36) + (1/log₃ 36) = 1. Find its value. 🧮", "options": {"A": "0", "B": "1", "C": "log 6", "D": "2"}, "correct_answer": "B"},
    {"id": 50, "difficulty": "hard", "question": "If log x = (log y)/2 = (log z)/3, find x⁴ ÷ (y²z²). ⚡", "options": {"A": "1", "B": "10", "C": "0", "D": "x"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "Simplify: (log 5)³ + (log 2)³ + 3 log 5 log 2 (log 7) / (log 10). 🧊", "options": {"A": "log 7", "B": "1", "C": "log 70", "D": "log 5 + log 2"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "hard", "question": "Given log (x² + y²) = log 2 + log x + log y, find (x/y) + (y/x). 🏹", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 53, "difficulty": "hard", "question": "If log (a+b)/3 = (1/2)(log a + log b), find a/b + b/a. 🤝", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "C"},
    {"id": 54, "difficulty": "hard", "question": "Solve: log₂ (x-1) + log₂ (x+3) = 3 + log₂ 3. ⏳", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "A"},
    {"id": 55, "difficulty": "hard", "question": "Prove that log sinθ + log cosθ + log secθ + log cosecθ = 0. Find its value for θ=45°. ✅", "options": {"A": "0", "B": "1", "C": "log 1", "D": "Both A & C"}, "correct_answer": "D"},
    {"id": 56, "difficulty": "hard", "question": "If log₂ [log₃ (log₄ x)] = 0, find x. 🎯", "options": {"A": "64", "B": "81", "C": "256", "D": "16"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "Simplify: log (√27)/log 9 × log 4/log ∛2. 🌳", "options": {"A": "3", "B": "6", "C": "9", "D": "1"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "Given log 2=0.3010, log 3=0.4771, find log (∛0.00048). 📍", "options": {"A": "-1.2672", "B": "-1.9343", "C": "-2.4343", "D": "-0.7672"}, "correct_answer": "B"},
    {"id": 59, "difficulty": "hard", "question": "If a = log 2, b = log 3, express log (1.2) + log (0.075) in terms of a and b. ✍️", "options": {"A": "a + b - 2", "B": "2a + b - 3", "C": "3a + 2b - 4", "D": "a - 2b - 1"}, "correct_answer": "B"},
    {"id": 60, "difficulty": "hard", "question": "Prove: (1/(1+log_b a)) + (1/(1+log_a b)) = 1. Find its numeric value. 🪐", "options": {"A": "0", "B": "1", "C": "log ab", "D": "2"}, "correct_answer": "B"}
  ]
}