Exponents and Logarithms
High-Yield Summary
- Six exponent rules: zero (X⁰=1), product (Xᵃ·Xᵇ=Xᵃ⁺ᵇ), quotient (Xᵃ/Xᵇ=Xᵃ⁻ᵇ), power ((Xᵃ)ᵇ=Xᵃᵇ), power of a quotient ((X/Y)ᵃ=Xᵃ/Yᵃ), negative (X⁻ᵃ=1/Xᵃ), fractional (Xᵃ/ᵇ=ᵇ√Xᵃ).
- Estimate square roots two ways: bound between nearby perfect squares (√50 just over 7, since 49<50<64), or factor out perfect squares (√180 = 2√45 = 6√5).
- Logarithm is the inverse of an exponent — 'what power gives this value?' Five rules: log(1)=0, logₐ(a)=1, log(AB)=logA+logB, log(A/B)=logA−logB, log(Aᴮ)=B·logA.
- Common log = base 10; natural log ln(X) = base e (≈2.718); convert via log₁₀(X) ≈ ln(X)/2.303.
- Quick estimate: log(n×10^m) ≈ m + 0.n, since log(n) always falls between 0 and 1 for a scientific-notation coefficient (1≤n<10).
The Six Exponent Rules
| Rule | Form |
|---|---|
| Zero exponent | X⁰ = 1 (X ≠ 0) |
| Product rule | Xᵃ · Xᵇ = Xᵃ⁺ᵇ |
| Quotient rule | Xᵃ / Xᵇ = Xᵃ⁻ᵇ |
| Power rule | (Xᵃ)ᵇ = Xᵃᵇ |
| Power of a quotient | (X/Y)ᵃ = Xᵃ/Yᵃ |
| Negative exponent | X⁻ᵃ = 1/Xᵃ |
| Fractional exponent | Xᵃ/ᵇ = ᵇ√(Xᵃ) |
The Five Logarithm Rules
| Rule | Form |
|---|---|
| Log of 1 | logₐ(1) = 0 |
| Log of the base | logₐ(a) = 1 |
| Product rule | log(AB) = log(A) + log(B) |
| Quotient rule | log(A/B) = log(A) − log(B) |
| Power rule | log(Aᴮ) = B · log(A) |
| Reciprocal rule | log(1/A) = −log(A) |
Estimating Square Roots by Perfect-Square Factoring
- 1Find a perfect-square factor of the number (e.g., 180 = 4 × 45).
- 2Pull the perfect square out of the radical: √180 = √4 · √45 = 2√45.
- 3Check if the remaining radicand has its own perfect-square factor (45 = 9 × 5) and repeat: √45 = 3√5.
- 4Combine: √180 = 2 · 3√5 = 6√5.
Common MCAT Trap
- log(n × 10^m) ≈ m + 0.n only works because n is between 1 and 10 (scientific-notation form) — don't apply this shortcut to an arbitrary coefficient outside that range.
- Don't confuse the power rule for exponents ((Xᵃ)ᵇ = Xᵃᵇ) with the power rule for logs (log(Aᴮ) = B·logA) — one multiplies exponents, the other pulls an exponent out as a multiplier.
Quick Recall
Simplify √50 without a calculator.
Estimate log(3.2 × 10⁵).