Definition: logₐ(x) = y  ⟺  aʸ = x

Product Rule:
  logₐ(xy) = logₐ(x) + logₐ(y)
  log(2×5) = log(2)+log(5) = log(10) = 1

Quotient Rule:
  logₐ(x/y) = logₐ(x) − logₐ(y)
  log(100/10) = log(100)−log(10) = 2−1 = 1

Power Rule:
  logₐ(xⁿ) = n × logₐ(x)
  log(10³) = 3 × log(10) = 3

Identities:
  logₐ(1) = 0        (any base)
  logₐ(a) = 1        (any base)
  a^(logₐ(x)) = x    (inverse property)
  logₐ(aˣ) = x       (inverse property)

The product and quotient rules convert multiplication/division into addition/subtraction — the fundamental trick that made slide rules and log tables possible for centuries.

Change of Base Formula:
  logₐ(b) = logc(b) / logc(a)
  logₐ(b) = ln(b) / ln(a)
  logₐ(b) = log₁₀(b) / log₁₀(a)

Example: log₅(100)
  = log₁₀(100)/log₁₀(5)
  = 2/0.69897
  = 2.861

Converting between bases:
  log₂(x) = ln(x)/ln(2) = log₁₀(x)/0.30103
  ln(x) = log₁₀(x)/log₁₀(e)
        = log₁₀(x)/0.43429
        = log₁₀(x) × 2.30259

Useful constants:
  ln(2)  ≈ 0.6931 → doubling time
  ln(10) ≈ 2.3026 → log₁₀ ↔ ln conversion
  log₂(10) ≈ 3.3219 → bits per digit

Any calculator with log₁₀ or ln can compute any base logarithm using the change of base formula. This is how hardware calculators typically work internally.

Euler's number: e ≈ 2.718281828...
  e = lim(n→∞) (1 + 1/n)ⁿ
  e = 1 + 1/1! + 1/2! + 1/3! + ...

Derivatives:
  d/dx ln(x) = 1/x
  d/dx logₐ(x) = 1/(x·ln(a))
  d/dx eˣ = eˣ  (unique self-derivative)

Integral:
  ∫(1/x)dx = ln|x| + C
  ∫ln(x)dx = x·ln(x) − x + C

Continuous growth/decay:
  A = A₀ × eʳᵗ
  Half-life: t½ = ln(2)/λ ≈ 0.693/λ
  Doubling time: t₂ = ln(2)/r ≈ 0.693/r
  Rule of 72: t₂ ≈ 72/r%
    At 6% interest: 72/6 = 12 years

The natural logarithm is 'natural' because its derivative is simply 1/x — the simplest possible form. This makes ln the default choice in calculus and physics.

Solving log equations:

1) logₐ(x) = c  →  x = aᶜ
   log₃(x) = 4  →  x = 3⁴ = 81

2) log(x) + log(x−3) = 1
   log(x(x−3)) = 1
   x(x−3) = 10
   x²−3x−10 = 0
   (x−5)(x+2) = 0
   x = 5  (x = −2 rejected: log domain)

3) Exponential equation:
   2ˣ = 100
   x·log(2) = log(100)
   x = 2/0.30103
   x = 6.644

4) Natural log equation:
   eˣ = 50
   x = ln(50)
   x ≈ 3.912

Domain: logₐ(x) defined only for x > 0
Always check solutions against domain!

When solving logarithmic equations, always verify that solutions don't produce log of zero or negative numbers — these are domain violations that must be rejected.