Exponents, Logarithms, and Financial Literacy
This domain combines exponent rules, logarithm laws, and the finance formulas for simple interest, compound interest, continuous growth, and doubling time.
Exponent rules
- aᵐ · aⁿ = aᵐ⁺ⁿ; aᵐ / aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; (ab)ⁿ = aⁿbⁿ.
- a⁰ = 1 for a ≠ 0; a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a; a^(m/n) = ⁿ√(aᵐ).
- Scientific notation writes numbers as c × 10ⁿ with 1 ≤ |c| < 10.
Logarithms
- Definition: log_b x = y exactly when bʸ = x. The log of zero or a negative number is undefined; log_b 1 = 0 and log_b b = 1.
- Product, quotient, and power laws: log(xy) = log x + log y; log(x/y) = log x − log y; log(xᵖ) = p log x.
- Change of base: log_b x = ln x / ln b. Use logs whenever the unknown sits in an exponent.
Financial formulas
- Simple interest: I = Prt, so A = P(1 + rt).
- Compound interest: A = P(1 + r/n)^(nt), with n compounding periods per year.
- Continuous compounding: A = Pe^(rt).
- APY measures the true annual yield once compounding is included; more frequent compounding raises it.
Worked example
$2,000 at 6% compounded monthly for 3 years.
- Periodic rate r/n = 0.06/12 = 0.005; periods nt = 12 · 3 = 36.
- A = 2000(1.005)³⁶ ≈ 2000(1.19668) = $2,393.36, so interest earned is about $393.36.
- Simple interest would give 2000(0.06)(3) = $360, so compounding adds roughly $33.
- Doubling time: solve 2 = (1.005)^(12t), so t = ln 2 / (12 ln 1.005) ≈ 0.6931 / 0.05985 ≈ 11.6 years.
Traps that cost marks
- Substituting the annual rate r where r/n belongs, or n where nt belongs.
- Mixing months and years without converting.
- Rounding intermediate values and drifting from the exact answer.
- Confusing log base 10 with natural log; either works if you use change of base consistently.
- Writing −3² when (−3)² was intended.