Exponential and Logarithmic Functions
Core ideas
Understand y = eˣ and y = ln x as inverses, with domains and asymptotes: eˣ > 0 for all x, and ln x only for x > 0. Apply log laws: log(ab) = log a + log b, log(a/b) = log a − log b, log(aⁿ) = n log a, and change of base. Solve exponential equations by taking logs of both sides, and log equations by combining into a single log then exponentiating, checking for extraneous solutions. Differentiate e^{kx} and ln(kx), and integrate 1/x to ln|x|. Model growth and decay with A = A₀e^{kt}, interpreting k and half-life.
Traps
- Writing ln(a + b) = ln a + ln b, which is false.
- Forgetting to reject solutions outside the domain of the original log equation.
- Losing the negative branch when solving x² = e^c type equations.
- Confusing gradient of eˣ with that of xⁿ.
- Misreading half-life: k is negative for decay.
Worked example
Solve e^{2x} − 5eˣ + 6 = 0. Let u = eˣ, so u² − 5u + 6 = 0 gives (u − 2)(u − 3) = 0. Thus eˣ = 2 or 3, and x = ln2 or ln3. Both are valid because eˣ is always positive. Rule of thumb: substitute u = eˣ to turn hidden quadratics into ordinary quadratics, then reject any u ≤ 0.