Exponents and Logarithms

Exponent laws

aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(m/n) = (ⁿ√a)ᵐ.

Logarithms

log_a x = y ⟺ aʸ = x (a > 0, a ≠ 1, x > 0).

• log₁₀ x is written log x; logₑ x is written ln x.
• log_a 1 = 0, log_a a = 1.

Laws of logarithms

Law	Example
log_a(xy) = log_a x + log_a y	log 20 + log 5 = log 100 = 2
log_a(x/y) = log_a x − log_a y	ln 12 − ln 3 = ln 4
log_a(xᵏ) = k log_a x	log₂ 8³ = 3 × 3 = 9

Change of base

log_a x = log_b x / log_b a, for example log₂ 10 = ln 10 / ln 2 ≈ 3.32.

Solving equations

• Same base: 2ˣ⁺¹ = 32 = 2⁵, so x = 4.
• Take logs: 3ˣ = 20, so x = ln 20 / ln 3 ≈ 2.73.
• Log equations: log₂(x + 3) + log₂(x − 3) = 4 → (x + 3)(x − 3) = 16 → x² = 25, x = 5 (x = −5 is rejected because logs need positive arguments).

Exponential models

e appears in continuous growth: A = A₀e^(kt).

Quick summary

• log_a x = y means aʸ = x.
• Product, quotient and power laws; change of base for calculators.
• Check solutions of log equations for positive arguments.