{
 "source": "https://peabody.kids/flashcards/math-log-rules",
 "id": "math-log-rules",
 "title": "Logarithm Rules and Values",
 "category": "math",
 "subject": "Math",
 "levels": [
  "9-12",
  "college"
 ],
 "description": "What a logarithm means, the laws of logarithms and common log values.",
 "tags": [
  "logarithms",
  "exponentials",
  "algebra"
 ],
 "frontLabel": "Rule",
 "backLabel": "Result",
 "cards": [
  [
   "Definition of a logarithm",
   "log_b(x) = y means bʸ = x"
  ],
  [
   "Product rule",
   "log_b(xy) = log_b x + log_b y"
  ],
  [
   "Quotient rule",
   "log_b(x/y) = log_b x − log_b y"
  ],
  [
   "Power rule",
   "log_b(xⁿ) = n log_b x"
  ],
  [
   "Change of base",
   "log_b x = log x / log b (any base)"
  ],
  [
   "Log of 1",
   "log_b 1 = 0"
  ],
  [
   "Log of the base",
   "log_b b = 1"
  ],
  [
   "Inverse property 1",
   "b^(log_b x) = x"
  ],
  [
   "Inverse property 2",
   "log_b(bˣ) = x"
  ],
  [
   "Common logarithm",
   "Base 10, written log x"
  ],
  [
   "Natural logarithm",
   "Base e, written ln x"
  ],
  [
   "Binary logarithm",
   "Base 2, written log₂ x"
  ],
  [
   "Domain of log x",
   "x > 0"
  ],
  [
   "log of a negative number",
   "Not defined among the real numbers"
  ],
  [
   "log 100",
   "2"
  ],
  [
   "log 1,000",
   "3"
  ],
  [
   "log 0.1",
   "−1"
  ],
  [
   "log 0.01",
   "−2"
  ],
  [
   "log 10",
   "1"
  ],
  [
   "log₂ 8",
   "3"
  ],
  [
   "log₂ 64",
   "6"
  ],
  [
   "log₂ 1024",
   "10"
  ],
  [
   "log₃ 81",
   "4"
  ],
  [
   "log₅ 125",
   "3"
  ],
  [
   "log₄ 2",
   "1/2"
  ],
  [
   "log₂ (1/8)",
   "−3"
  ],
  [
   "ln e",
   "1"
  ],
  [
   "ln 1",
   "0"
  ],
  [
   "ln(e²)",
   "2"
  ],
  [
   "e^(ln 5)",
   "5"
  ],
  [
   "Expand log(ab/c)",
   "log a + log b − log c"
  ],
  [
   "Condense 2 log x + log y",
   "log(x²y)"
  ],
  [
   "Solve 2ˣ = 16",
   "x = 4"
  ],
  [
   "Solve 3ˣ = 5",
   "x = log 5 / log 3 (about 1.465)"
  ],
  [
   "Solve log₂ x = 5",
   "x = 32"
  ],
  [
   "Is log(x + y) = log x + log y?",
   "No. The product rule needs log(xy)"
  ],
  [
   "Is log(x)/log(y) the same as log(x/y)?",
   "No. They are different quantities"
  ],
  [
   "Graph of y = log x: vertical asymptote",
   "x = 0"
  ],
  [
   "Graph of y = log x: x-intercept",
   "x = 1"
  ],
  [
   "Value of e (to 5 decimals)",
   "2.71828"
  ],
  [
   "Half-life formula idea",
   "Amount = start × (1/2)^(t / half-life)"
  ],
  [
   "Decibel scale uses",
   "A logarithm of a power ratio (10 log₁₀)"
  ],
  [
   "Richter scale uses",
   "A base-10 logarithm of shaking amplitude"
  ],
  [
   "pH formula",
   "pH = −log₁₀[H⁺]"
  ]
 ]
}