Exponential and Logarithmic Functions: Growth, Decay, and Inverses
What you must understand
Exponential functions model constant-percentage change; logarithmic functions are their inverses. Key concepts: base restrictions (b > 0, b ≠ 1), domain and range, transformations, properties of logarithms, solving exponential and logarithmic equations, and modeling with exponential growth/decay and half-life or doubling time.
Traps that show up
- Domain of logs: log(x) requires x > 0. A transformed log like ln(x − 3) requires x > 3. Always state the domain before solving.
- Extraneous solutions: solving log equations can produce values outside the domain; check each candidate.
- Base-e versus base-10 confusion: ln is base e, log is base 10 (in most contexts). Conversion: log_b(x) = ln(x)/ln(b).
- Growth versus decay: b > 1 grows; 0 < b < 1 decays. The multiplier per period is 1 + r for growth and 1 − r for decay.
- Log properties: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a. These do not distribute over sums.
Rule of thumb
To solve b^x = k, take the log of both sides: x = log(k)/log(b). To solve log_b(x) = k, rewrite as x = b^k, then verify the domain.
Worked example
Solve 3^(2x) = 81. Since 81 = 3⁴, 2x = 4, so x = 2. Alternatively, 2x·ln3 = ln81 → x = ln81/(2 ln3) = 2.