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⁺]

Free flashcards from Peabody at peabody.kids/flashcards/math-log-rules