Kung Fu Math · kungfumath.app
Logarithms, powers & roots
A logarithm answers: what exponent produces this number? In logᵦ x, require b > 0, b ≠ 1 and x > 0.
Power rules
- aᵐ × aⁿ = aᵐ⁺ⁿ
- aᵐ / aⁿ = aᵐ⁻ⁿ (a ≠ 0)
- (aᵐ)ⁿ = aᵐⁿ
- (ab)ⁿ = aⁿbⁿ
- a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ
- √a = a¹ᐟ² (a ≥ 0); √(a²) = |a|
Logarithm rules · x, y > 0
- logᵦ(xy) = logᵦ x + logᵦ y
- logᵦ(x/y) = logᵦ x − logᵦ y
- logᵦ(xʳ) = r logᵦ x
- logᵦ x = ln x / ln b
- logᵦ 1 = 0; logᵦ b = 1
- ln(eˣ) = x; eˡⁿˣ = x (x > 0)
Examples & traps
- 2³ × 2⁴ = 2⁷ = 128
- log₁₀ 1000 = 3; ln e² = 2
- √144 = 12; √(9 + 16) = 5, not 7
- log(x + y) does NOT equal log x + log y.
Kung Fu Math · logarithms-powers · kungfumath.app/cheat-sheets