{
  "title": "Applications of Logarithms - Grade 9 ICSE",
  "total_questions": 60,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Logarithms are used to simplify calculations involving large numbers. What is log 1000? 🧮", "options": {"A": "2", "B": "3", "C": "4", "D": "10"}, "correct_answer": "B"},
    {"id": 2, "difficulty": "easy", "question": "If the population of bacteria doubles every hour, and we start with 100, after 3 hours, the population is 100 × 2³. What is log₂ of the final population? 🦠", "options": {"A": "log₂ 100 + 3", "B": "100 + 3", "C": "log₂ 800", "D": "Both A & C"}, "correct_answer": "D"},
    {"id": 3, "difficulty": "easy", "question": "The Richter scale for earthquakes uses logarithms. If one quake is 10 times stronger than another, how much greater is its Richter number? 🌍", "options": {"A": "1", "B": "10", "C": "log 10", "D": "2"}, "correct_answer": "A"},
    {"id": 4, "difficulty": "easy", "question": "Sound intensity is measured in decibels (dB), using log₁₀. If sound A is 100 times more intense than sound B, how many dB greater is it? 🔊", "options": {"A": "10 dB", "B": "20 dB", "C": "100 dB", "D": "1 dB"}, "correct_answer": "B"},
    {"id": 5, "difficulty": "easy", "question": "pH of a solution is -log₁₀[H⁺]. If [H⁺] = 0.001, what is the pH? 🧪", "options": {"A": "1", "B": "2", "C": "3", "D": "4"}, "correct_answer": "C"},
    {"id": 6, "difficulty": "easy", "question": "In compound interest formula A = P(1 + r)ⁿ, to find n, we often use logarithms. If A=2P, and r=0.1, then (1.1)ⁿ = 2. What is n log (1.1) equal to? 💰", "options": {"A": "log 2", "B": "log 1.1", "C": "log P", "D": "2 log P"}, "correct_answer": "A"},
    {"id": 7, "difficulty": "easy", "question": "If log 2 ≈ 0.3010, how many digits are in 2¹⁰? 🔢", "options": {"A": "3", "B": "4", "C": "5", "D": "10"}, "correct_answer": "B"},
    {"id": 8, "difficulty": "easy", "question": "The magnitude of a star is given by m = -2.5 log₁₀ (brightness). If a star is 100× brighter, how much does its magnitude change? ⭐", "options": {"A": "-5", "B": "-2.5", "C": "+5", "D": "+2.5"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "To solve 10ˣ = 50, we use log. So x = ? 📝", "options": {"A": "log 50", "B": "log 5", "C": "log 10", "D": "50 log 10"}, "correct_answer": "A"},
    {"id": 10, "difficulty": "easy", "question": "Logarithms can turn multiplication into addition. To find 123 × 456 using logs, we compute: log(123×456) = ? ➕", "options": {"A": "log 123 + log 456", "B": "log 123 × log 456", "C": "log 579", "D": "123 + 456"}, "correct_answer": "A"},
    {"id": 11, "difficulty": "easy", "question": "If a radioactive substance halves every 10 years (half-life), after 30 years, what fraction remains? Use logs if needed. ⚛️", "options": {"A": "1/2", "B": "1/4", "C": "1/8", "D": "1/16"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "Logarithms are used in measuring acidity. pH = -log[H⁺]. If pH = 7, what is [H⁺]? 💧", "options": {"A": "10⁻⁷", "B": "7", "C": "0.7", "D": "10⁷"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "In music, the number of octaves between frequencies f₁ and f₂ is log₂(f₂/f₁). If f₂ = 4f₁, how many octaves? 🎵", "options": {"A": "1", "B": "2", "C": "4", "D": "log₂ 4"}, "correct_answer": "B"},
    {"id": 14, "difficulty": "easy", "question": "To find how long an investment doubles at 5% p.a., we solve (1.05)ⁿ = 2. Using log, n = ? 📈", "options": {"A": "log 2 / log 1.05", "B": "log 1.05 / log 2", "C": "2 log 1.05", "D": "log 2.05"}, "correct_answer": "A"},
    {"id": 15, "difficulty": "easy", "question": "Logarithms help solve exponential equations. Solve 3ˣ = 81. What is x? ⚡", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "C"},
    {"id": 16, "difficulty": "easy", "question": "The decibel gain in an amplifier is 10 log₁₀(P_out/P_in). If power doubles, gain is? 🔊", "options": {"A": "10 log 2 dB", "B": "2 dB", "C": "10 dB", "D": "3 dB"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "If log N = 5.3010, how many digits are in N? (log 2≈0.3010) 🔍", "options": {"A": "5", "B": "6", "C": "7", "D": "8"}, "correct_answer": "B"},
    {"id": 18, "difficulty": "easy", "question": "In computer science, logarithms measure algorithm efficiency. log₂ 1024 = ? 💻", "options": {"A": "8", "B": "10", "C": "12", "D": "1024"}, "correct_answer": "B"},
    {"id": 19, "difficulty": "easy", "question": "If the intensity of light decreases by a factor of 100, what is the change in magnitude? (m = -2.5 log I) 🌞", "options": {"A": "+5", "B": "-5", "C": "+2.5", "D": "-2.5"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Solve using logs: 5ˣ = 25. What is x? ✅", "options": {"A": "1", "B": "2", "C": "5", "D": "25"}, "correct_answer": "B"},
    {"id": 21, "difficulty": "medium", "question": "Given log 2 = 0.3010, log 3 = 0.4771, find the number of digits in 6¹⁰. 📏", "options": {"A": "7", "B": "8", "C": "9", "D": "10"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "A sound of 70 dB is how many times more intense than 40 dB? (I ∝ 10^(dB/10)) 🔊", "options": {"A": "3 times", "B": "30 times", "C": "100 times", "D": "1000 times"}, "correct_answer": "D"},
    {"id": 23, "difficulty": "medium", "question": "The population P after t years is P = P₀ e^(kt). To find t when P doubles, we solve e^(kt)=2. What is t? 🌱", "options": {"A": "ln 2 / k", "B": "k / ln 2", "C": "2/k", "D": "log 2 / k"}, "correct_answer": "A"},
    {"id": 24, "difficulty": "medium", "question": "The formula for the magnitude difference between two stars is m₁ - m₂ = -2.5 log (b₁/b₂). If m₁ - m₂ = 5, what is b₁/b₂? ✨", "options": {"A": "100", "B": "1/100", "C": "10", "D": "1/10"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "If log₁₀ x = -2.5, then x = ? 📉", "options": {"A": "0.00316", "B": "0.0316", "C": "0.0025", "D": "0.00025"}, "correct_answer": "A"},
    {"id": 26, "difficulty": "medium", "question": "An earthquake of magnitude 6 is how many times stronger than magnitude 4? (Energy ∝ 10^(1.5M)) 🌋", "options": {"A": "1.5 times", "B": "10 times", "C": "100 times", "D": "1000 times"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "Solve for t: 200 = 50 × 2^(t/3). ⏱️", "options": {"A": "3", "B": "6", "C": "9", "D": "12"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "If log (x² - 4) = log 5 + log (x-2), find x. 🧩", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 29, "difficulty": "medium", "question": "The half-life of carbon-14 is 5730 years. After how many years will 1/16 remain? Use logs. 🦴", "options": {"A": 11460, "B": 17190, "C": 22920, "D": 28650}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "Given log 5 = 0.6990, find log 0.04. 📊", "options": {"A": "-1.3010", "B": "-1.3980", "C": "-1.6990", "D": "-2.3010"}, "correct_answer": "B"},
    {"id": 31, "difficulty": "medium", "question": "A bank offers 8% compound interest. How many years to triple the money? (Use log 3≈0.4771, log 1.08≈0.0334) 🏦", "options": {"A": "≈14.2", "B": "≈15.1", "C": "≈16.8", "D": "≈18.5"}, "correct_answer": "A"},
    {"id": 32, "difficulty": "medium", "question": "If log₃ (x+5) - log₃ (x-1) = 1, find x. 🔎", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "The loudness L in phons is L = 10 log₁₀(I/I₀). If I increases by a factor 10⁶, L increases by? 🔊", "options": {"A": "6 dB", "B": "60 dB", "C": "10⁶ dB", "D": "log 10⁶ dB"}, "correct_answer": "B"},
    {"id": 34, "difficulty": "medium", "question": "Solve: 4^(x+1) = 8^(x-1). ⚡", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "C"},
    {"id": 35, "difficulty": "medium", "question": "Given log 2=0.3010, find log (0.0625). 📍", "options": {"A": "-1.2040", "B": "-1.3010", "C": "-2.2040", "D": "-2.3010"}, "correct_answer": "A"},
    {"id": 36, "difficulty": "medium", "question": "If log (a² + b²) = log 2 + log a + log b, find (a-b)². 🧮", "options": {"A": "0", "B": "a²+b²", "C": "2ab", "D": "ab"}, "correct_answer": "A"},
    {"id": 37, "difficulty": "medium", "question": "The noise level in a room increases from 30 dB to 60 dB. How many times more intense? 🏠", "options": {"A": "2 times", "B": "10 times", "C": "100 times", "D": "1000 times"}, "correct_answer": "D"},
    {"id": 38, "difficulty": "medium", "question": "How many years for ₹5000 at 10% compounded annually to become ₹6655? (log 1.331=0.1242, log 1.1=0.0414) 💵", "options": {"A": "2", "B": "3", "C": "4", "D": "5"}, "correct_answer": "B"},
    {"id": 39, "difficulty": "medium", "question": "If log (x+y) = log x + log y, find y in terms of x. 🤔", "options": {"A": "y = x/(x-1)", "B": "y = x-1", "C": "y = x+1", "D": "y = 1/x"}, "correct_answer": "A"},
    {"id": 40, "difficulty": "medium", "question": "The formula for the number of bacteria N after t minutes is N = 1000 × 2^(t/20). When will N = 32000? 🦠", "options": {"A": "60 min", "B": "80 min", "C": "100 min", "D": "120 min"}, "correct_answer": "C"},
    {"id": 41, "difficulty": "hard", "question": "Given log 2=0.3010, log 3=0.4771, find the number of digits in (12)^15. 🔢", "options": {"A": "15", "B": "16", "C": "17", "D": "18"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "hard", "question": "Solve: log₄ (x) + log₈ (x) = 8/3. ⚔️", "options": {"A": "16", "B": "32", "C": "64", "D": "128"}, "correct_answer": "C"},
    {"id": 43, "difficulty": "hard", "question": "A radioactive element decays to 1/√2 of original in 5 days. What is the half-life? (Use log 2=0.3010) ⚛️", "options": {"A": "≈12.2 days", "B": "≈10 days", "C": "≈15 days", "D": "≈8.3 days"}, "correct_answer": "A"},
    {"id": 44, "difficulty": "hard", "question": "If log (x² + y²) = 2 log (x+y), prove x=y. Then if x+y=10, find x. ✅", "options": {"A": "2", "B": "4", "C": "5", "D": "6"}, "correct_answer": "C"},
    {"id": 45, "difficulty": "hard", "question": "An earthquake of magnitude 8 releases how many times more energy than magnitude 5? (E ∝ 10^(1.5M)) 🌋", "options": {"A": "30 times", "B": "100 times", "C": "1000 times", "D": "31,623 times"}, "correct_answer": "D"},
    {"id": 46, "difficulty": "hard", "question": "Given log 5 = 0.6990, find log (√(0.04)) + log (∛(0.008)). 🌳", "options": {"A": "-1", "B": "-2", "C": "-3", "D": "0"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "hard", "question": "If log (sin x) + log (cos x) + log (tan x) + log (cot x) = 0, find x for 0°<x<90°. 📐", "options": {"A": "30°", "B": "45°", "C": "60°", "D": "any angle"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "hard", "question": "Solve: 2^(2x) - 6 × 2^x + 8 = 0. 🔥", "options": {"A": "1, 2", "B": "0, 3", "C": "1, 3", "D": "2, 3"}, "correct_answer": "A"},
    {"id": 49, "difficulty": "hard", "question": "The pH of a solution changes from 3 to 5. By what factor does [H⁺] change? 🧪", "options": {"A": "2 times", "B": "10 times", "C": "100 times", "D": "0.01 times"}, "correct_answer": "D"},
    {"id": 50, "difficulty": "hard", "question": "If log₃ 2 = a, log₅ 2 = b, express log₆₀ 2 in terms of a and b. 🔗", "options": {"A": "ab/(a+b+1)", "B": "a+b/(1+ab)", "C": "(a+b)/(1+ab)", "D": "1/(a+b+ab)"}, "correct_answer": "A"},
    {"id": 51, "difficulty": "hard", "question": "A sum of money doubles in 6 years at compound interest. In how many years will it be 8 times? 📅", "options": {"A": "12", "B": "18", "C": "24", "D": "36"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "hard", "question": "Solve: log (x+1) - log (x-1) = 2 log 2 - log (x-2). 🌀", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "hard", "question": "Given log₁₀ 2=0.3010, find the smallest integer n such that 2^n > 10¹⁰. ⏳", "options": {"A": "33", "B": "34", "C": "35", "D": "36"}, "correct_answer": "B"},
    {"id": 54, "difficulty": "hard", "question": "The intensity of light decreases as 1/d². If magnitude m = -2.5 log I, how does m change with distance d? 🌟", "options": {"A": "m ∝ log d", "B": "m ∝ 2 log d", "C": "m ∝ 5 log d", "D": "m ∝ -2.5 log d"}, "correct_answer": "C"},
    {"id": 55, "difficulty": "hard", "question": "If log (x² + y²) = log 2 + log x + log y, find log (x+y). 🧬", "options": {"A": "log x + log y", "B": "(1/2)(log x + log y + log 2)", "C": "log 2 + (1/2)(log x + log y)", "D": "log (2xy)"}, "correct_answer": "B"},
    {"id": 56, "difficulty": "hard", "question": "The population of a town grows at 5% p.a. When will it triple? (log 3=0.4771, log 1.05=0.0212) 🏘️", "options": {"A": "≈22.5 years", "B": "≈25.1 years", "C": "≈30.2 years", "D": "≈35.3 years"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "hard", "question": "Prove: (log a / log b) × (log b / log c) × (log c / log a) = 1. If a=2, b=4, c=16, verify. ✅", "options": {"A": "True, value is 1", "B": "False", "C": "True only if a=b=c", "D": "Cannot be determined"}, "correct_answer": "A"},
    {"id": 58, "difficulty": "hard", "question": "Solve: log₂ (x+1) + log₂ (x+3) = 3 + log₂ 3. 📏", "options": {"A": "3", "B": "4", "C": "5", "D": "6"}, "correct_answer": "A"},
    {"id": 59, "difficulty": "hard", "question": "If logₐ b = 2, log_b c = 3, log_c d = 4, find log_a d. 🔗", "options": {"A": "6", "B": "9", "C": "12", "D": "24"}, "correct_answer": "D"},
    {"id": 60, "difficulty": "hard", "question": "A certain bank's interest makes money increase by 50% in 4 years. How long to double? (Use log 2=0.3010, log 1.5=0.1761) 💰", "options": {"A": "≈6.4 years", "B": "≈6.8 years", "C": "≈7.2 years", "D": "≈8.0 years"}, "correct_answer": "B"}
  ]
}