SL 1.5—Integer exponents and logarithms
- Syllabus
- First assessment 2021
- Objective
- —
- Level
- SL
Exponent laws and logarithms undo one another
Integer exponents describe repeated multiplication and its inverse operations: aᵐaⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ, and a⁰ = 1 for a ≠ 0. A logarithm reverses exponentiation: log_b(x) = y exactly when bʸ = x.
To solve 3·2ˣ = 24, divide by 3 to get 2ˣ = 8 and hence x = 3. For a non-integer result, take logarithms: x = ln(8/3)/ln 2.
The logarithm requires b > 0, b ≠ 1 and x > 0. Apply exponent laws only to products, quotients and powers; log(a + b) is not log a + log b.