{
  "title": "Definition and Properties of Logarithms - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "If aˣ = N, then the logarithm form is? 📝", "options": {"A": "x = logₐ N", "B": "a = logₓ N", "C": "N = logₐ x", "D": "x = log_N a"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "What is the base in log₁₀ 100 = 2? 🔢", "options": {"A": "2", "B": "10", "C": "100", "D": "log"}, "correct_answer": "B"},
    {"id": 3, "difficulty": "easy", "question": "If log₂ 8 = 3, what is the exponential form? 🔄", "options": {"A": "2³ = 8", "B": "3² = 8", "C": "8² = 3", "D": "2⁸ = 3"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "logₐ 1 is always equal to? 😊", "options": {"A": "a", "B": "1", "C": "0", "D": "10"}, "correct_answer": "C"},
    {"id": 5, "difficulty": "easy", "question": "What is logₐ a? ✅", "options": {"A": "0", "B": "1", "C": "a", "D": "10"}, "correct_answer": "B"},
    {"id": 6, "difficulty": "easy", "question": "If logₓ 16 = 2, what is x? 🎯", "options": {"A": "2", "B": "4", "C": "8", "D": "16"}, "correct_answer": "B"},
    {"id": 7, "difficulty": "easy", "question": "What is the value of log₅ 25? ✏️", "options": {"A": "1", "B": "2", "C": "5", "D": "25"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "The logarithm of a number to the same base is? 🔁", "options": {"A": "0", "B": "1", "C": "the number itself", "D": "10"}, "correct_answer": "B"},
    {"id": 9, "difficulty": "easy", "question": "Convert to log form: 10³ = 1000. 🔟", "options": {"A": "log 3 = 1000", "B": "log₁₀ 1000 = 3", "C": "log 1000 = 10", "D": "log 1000 = 3"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "If log₃ x = 4, what is x? 📈", "options": {"A": "12", "B": "64", "C": "81", "D": "7"}, "correct_answer": "C"},
    {"id": 11, "difficulty": "easy", "question": "What is the base of common logarithms? 🌍", "options": {"A": "e", "B": "2", "C": "10", "D": "1"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "log 1 (base 10) is? 🧮", "options": {"A": "0", "B": "1", "C": "10", "D": "undefined"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "If 5² = 25, then log₅ 25 = ? 🔄", "options": {"A": "2", "B": "5", "C": "25", "D": "1/2"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Which is correct? logₐ (m×n) = ? 🤔", "options": {"A": "logₐ m × logₐ n", "B": "logₐ m + logₐ n", "C": "logₐ m - logₐ n", "D": "logₐ m ÷ logₐ n"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "What is log_b a × log_a b equal to? 🔄", "options": {"A": "0", "B": "1", "C": "log ab", "D": "a×b"}, "correct_answer": "B"},
    {"id": 16, "difficulty": "easy", "question": "If log N = 2, what is N? (common log) 📊", "options": {"A": "20", "B": "100", "C": "2", "D": "0.1"}, "correct_answer": "B"},
    {"id": 17, "difficulty": "easy", "question": "What is log₄ 1? 🎈", "options": {"A": "4", "B": "1", "C": "0", "D": "¼"}, "correct_answer": "C"},
    {"id": 18, "difficulty": "easy", "question": "The statement 'logₐ a = 1' is true for? ✅", "options": {"A": "a=1 only", "B": "a>0, a≠1", "C": "all a", "D": "a=10 only"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "If logₓ 1 = 0, what can x be? 🌟", "options": {"A": "any positive number ≠1", "B": "only 10", "C": "only 1", "D": "0"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "What is the value of log₁₀ 0.01? 📉", "options": {"A": "1", "B": "2", "C": "-1", "D": "-2"}, "correct_answer": "D"},
    {"id": 21, "difficulty": "medium", "question": "If log (x² - 3x + 3) = 0, find x. ⚖️", "options": {"A": "1", "B": "2", "C": "1 or 2", "D": "0"}, "correct_answer": "C"},
    {"id": 22, "difficulty": "medium", "question": "What is the condition for logₐ b to be defined? 🛡️", "options": {"A": "a>0, b>0, a≠1", "B": "a>0, b≥0", "C": "a>1, b>0", "D": "a≠1, b any real"}, "correct_answer": "A"},
    {"id": 23, "difficulty": "medium", "question": "If logₓ (1/8) = -3, then x = ? 🧊", "options": {"A": "2", "B": "-2", "C": "1/2", "D": "8"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "Prove: a^(logₐ N) = N. If a=2, N=8, what is the value? 🔥", "options": {"A": "3", "B": "8", "C": "16", "D": "log 8"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "Given log 2 = 0.3010, what is log 0.5? 📍", "options": {"A": "0.3010", "B": "-0.3010", "C": "0.6990", "D": "-0.6990"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "If log (x+5) = 1, find x (base 10). 🧩", "options": {"A": "5", "B": "10", "C": "15", "D": "0"}, "correct_answer": "A"},
    {"id": 27, "difficulty": "medium", "question": "What is log₄ 8? (Hint: write as power) 🌱", "options": {"A": "1/2", "B": "3/2", "C": "2/3", "D": "1.5"}, "correct_answer": "D"},
    {"id": 28, "difficulty": "medium", "question": "If log (m/n) = 0, what is true about m & n? ⚖️", "options": {"A": "m = n", "B": "m > n", "C": "m < n", "D": "m = 0"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "Find x if log₉ 27 = x. 📐", "options": {"A": "1/2", "B": "2/3", "C": "3/2", "D": "2"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "Which is greater: log₂ 3 or log₃ 2? 📊", "options": {"A": "log₂ 3", "B": "log₃ 2", "C": "equal", "D": "can't compare"}, "correct_answer": "A"},
    {"id": 31, "difficulty": "medium", "question": "If logₐ b = 2 and log_b c = 3, find logₐ c. 🔗", "options": {"A": "5", "B": "6", "C": "8", "D": "1"}, "correct_answer": "B"},
    {"id": 32, "difficulty": "medium", "question": "Simplify: log₅ (1/125). ❄️", "options": {"A": "3", "B": "-3", "C": "1/3", "D": "-1/3"}, "correct_answer": "B"},
    {"id": 33, "difficulty": "medium", "question": "Given log x = -2, find x (common log). 🔎", "options": {"A": "100", "B": "0.01", "C": "-0.01", "D": "0.1"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "If log₂ (log₃ (log₄ x)) = 0, find x. 🌀", "options": {"A": "64", "B": "81", "C": "256", "D": "16"}, "correct_answer": "A"},
    {"id": 35, "difficulty": "medium", "question": "What is the characteristic of log 0.003? 📏", "options": {"A": "0", "B": "-2", "C": "-3", "D": "3"}, "correct_answer": "C"},
    {"id": 36, "difficulty": "medium", "question": "If log (a² - b²) = log (a+b) + log (a-b), what condition is needed? ⚠️", "options": {"A": "a>b>0", "B": "a,b>0", "C": "a≠b", "D": "all of these"}, "correct_answer": "D"},
    {"id": 37, "difficulty": "medium", "question": "Solve: log₄ (x+3) = 1/2. ✨", "options": {"A": "1", "B": "-1", "C": "5", "D": "-5"}, "correct_answer": "B"},
    {"id": 38, "difficulty": "medium", "question": "If log 2 = p, log 3 = q, express log 12 in terms of p and q. 📝", "options": {"A": "p+q", "B": "2p+q", "C": "p+2q", "D": "p²+q"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "What is the value of log₀.₅ 4? 📉", "options": {"A": "2", "B": "-2", "C": "1/2", "D": "-1/2"}, "correct_answer": "B"},
    {"id": 40, "difficulty": "medium", "question": "If log₁₀ x = -1.5, then x = ? 🔬", "options": {"A": "0.015", "B": "0.0316", "C": "0.0015", "D": "0.316"}, "correct_answer": "B"},
    {"id": 41, "difficulty": "hard", "question": "Prove: (logₐ b)(log_b c)(log_c d) = logₐ d. If a=2, b=4, c=16, d=256, verify. ✅", "options": {"A": "log₂ 256 = 8", "B": "8", "C": "true, value is 8", "D": "All of these"}, "correct_answer": "D"},
    {"id": 42, "difficulty": "hard", "question": "If (log x)/(log y) = 2/3 and xy = 1000, find x and y. 🔎", "options": {"A": "x=100, y=10", "B": "x=10, y=100", "C": "x=∛1000, y=√1000", "D": "x=100^(2/3), y=100^(1/3)"}, "correct_answer": "A"},
    {"id": 43, "difficulty": "hard", "question": "Solve: log (log (log x)) = 0 (common logs). 🌌", "options": {"A": "10", "B": "100", "C": "10¹⁰", "D": "10¹⁰⁰"}, "correct_answer": "C"},
    {"id": 44, "difficulty": "hard", "question": "If log (a² + b²) = log 2 + log a + log b, find (a/b) + (b/a). 🏹", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "hard", "question": "Given log₁₀ 2 = 0.3010, how many digits are in 2⁵⁰? 🔢", "options": {"A": "15", "B": "16", "C": "17", "D": "18"}, "correct_answer": "B"},
    {"id": 46, "difficulty": "hard", "question": "If log (x+y)/5 = (1/2)(log x + log y), find x/y + y/x. 🤝", "options": {"A": "23", "B": "24", "C": "25", "D": "26"}, "correct_answer": "A"},
    {"id": 47, "difficulty": "hard", "question": "Prove: log (1+2+3) = log 1 + log 2 + log 3 is false. Why? ❌", "options": {"A": "LHS = log 6, RHS = log 6", "B": "LHS = log 6, RHS = 0", "C": "LHS ≠ RHS", "D": "log laws misapplied"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "hard", "question": "If log₂ [log₃ (log₂ x)] = 1, find x. 🧩", "options": {"A": 512, "B": 256, "C": 128, "D": 64}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "Solve: logₓ (2x-3) = 2. ⚔️", "options": {"A": "3", "B": "2", "C": "3 or 1", "D": "3 only"}, "correct_answer": "D"},
    {"id": 50, "difficulty": "hard", "question": "Given log (sinθ) + log (cosθ) + log (tanθ) = 0, find θ for 0°<θ<90°. 📐", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "90°"}, "correct_answer": "B"},
    {"id": 51, "difficulty": "hard", "question": "If log (x² + y²) = log x + log y + log 2, show x=y. Find x if x+y=10. ✅", "options": {"A": "2", "B": "5", "C": "√10", "D": "10"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "hard", "question": "Simplify: log (√(x+1) + √x) + log (√(x+1) - √x). 🌳", "options": {"A": "log 1", "B": "0", "C": "log (x+1 - x)", "D": "All of these"}, "correct_answer": "D"},
    {"id": 53, "difficulty": "hard", "question": "If log (a+b)/3 = (1/2)(log a + log b), prove a² + b² = 7ab. Then find a/b if a=2b. 🔄", "options": {"A": "2", "B": "1/2", "C": "√2", "D": "2 only"}, "correct_answer": "A"},
    {"id": 54, "difficulty": "hard", "question": "Evaluate: log₂ 3 × log₃ 4 × log₄ 5 × log₅ 6 × log₆ 7 × log₇ 8. ⛓️", "options": {"A": "1", "B": "2", "C": "3", "D": "log 8"}, "correct_answer": "C"},
    {"id": 55, "difficulty": "hard", "question": "Given log 2=0.3010, log 3=0.4771, find log (√0.96) to four decimals. 📍", "options": {"A": "-0.0088", "B": "-0.0177", "C": "0.0088", "D": "0.0177"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "hard", "question": "If x = 1+log a, y = 1+log b, z = 1+log c, and a,b,c in G.P., show 1/x, 1/y, 1/z are in H.P. ✅", "options": {"A": "True", "B": "False", "C": "Sometimes", "D": "Cannot determine"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "Solve: logₓ (x+12) - 2 logₓ 2 = 1. ⏳", "options": {"A": "2", "B": "4", "C": "6", "D": "8"}, "correct_answer": "B"},
    {"id": 58, "difficulty": "hard", "question": "If a, b, c are consecutive integers, prove log (1+ac) = 2 log b. Verify for a=1,b=2,c=3. 🧾", "options": {"A": "log 4 = 2 log 2", "B": "true", "C": "both A & B", "D": "false"}, "correct_answer": "C"},
    {"id": 59, "difficulty": "hard", "question": "Given log (√(1+x)+√x) = 1/2, find x. 🔑", "options": {"A": 0.9, "B": 0.99, "C": 0.09, "D": 0.009}, "correct_answer": "A"},
    {"id": 60, "difficulty": "hard", "question": "If log (x² + y²) = 2 log (x+y), prove x=y. Find x if x+y=8. ⚖️", "options": {"A": 2, "B": 4, "C": 6, "D": 8}, "correct_answer": "B"}
  ]
}