Logarithms and Exponential Functions
Core ideas
Candidates must use the laws of logarithms: log(xy) = log x + log y, log(x/y) = log x − log y, log(xⁿ) = n log x, and understand that logₐ a = 1 and logₐ 1 = 0. The relationship y = aˣ is equivalent to x = logₐ y. Graphs of exponential functions and their transformations must be known. Solving equations of the form aˣ = b uses logarithms; equations reducible to quadratics in aˣ are common. Also examinable: the natural exponential and logarithm, and straight-line forms used to determine constants from experimental data.
Traps
- Writing log(x + y) = log x + log y - this is false.
- Forgetting to check that logarithmic arguments are positive.
- Using log base 10 when base e is required, or vice versa.
- Losing the second root when solving a quadratic in aˣ, especially a negative root that must be rejected.
- Rounding intermediate values and losing accuracy.
Worked example
Solve 3^(2x) = 20. Taking logs: 2x log 3 = log 20, so x = log 20/(2 log 3) ≈ 1.3010/(0.9542) ≈ 1.363.
Rule of thumb
When the unknown is in an exponent, take logarithms of both sides and use the power law to bring it down.