Logarithmic and Exponential Functions (Copy)
6 Logarithmic & Exponential Functions – Quick Reference
| Topic | Key Facts / Properties | Examples | Common Pitfalls |
|---|---|---|---|
| Exponential Function | y = eⁿˣ → domain: x ∈ ℝ, range: y > 0 Always passes through (0, 1) Horizontal asymptote y = 0 Increasing if n > 0, decreasing if n < 0 | y = 2e³ˣ + 1 → Asymptote y = 1 | Forgetting to shift asymptote when +a present |
| Logarithmic Function | y = ln(x) → domain: x > 0, range: y ∈ ℝ Passes through (1, 0) Vertical asymptote x = 0 Inverse of eˣ | y = 3ln(2x – 4) → Asymptote x = 2 | Using ln(negative) → undefined |
| Inverse Relationship | If y = eˣ → x = ln(y) If y = aˣ → x = logₐ(y) | eˡⁿˣ = x (x > 0) ln(eᵏ) = k | Applying ln to negative numbers |
| Laws of Logarithms | logₐ(mn) = logₐm + logₐn logₐ(m/n) = logₐm – logₐn logₐ(mᵏ) = k logₐm Change of base: logₐm = log_bm / log_ba | log₅(25) = 2 log₂(8) = 3 | Wrongly splitting log of a sum: log(a + b) ≠ log a + log b |
| Converting Logs | logₑ(x) = ln(x) log₁₀(x) is common log | log₁₀(1000) = 3 ln(e⁴) = 4 | Mixing up bases |
| Solving Exponential Eqns | aˣ = b → take ln both sides → x ln(a) = ln(b) → solve for x | Solve 3e²ˣ = 27 → e²ˣ = 9 → 2x = ln 9 → x = (ln 9)/2 | Forgetting to isolate exponential before ln |
| Solving Logarithmic Eqns | logₐ(f(x)) = k → f(x) = aᵏ, then solve | log₂(x – 1) = 3 → x – 1 = 8 → x = 9 | Not checking x > 0 condition for arguments |
Quick Graph Notes:
- y = eⁿˣ + a → Horizontal asymptote y = a
- y = k ln(ax + b) → Vertical asymptote x = –b/a
- Sketches should clearly mark asymptotes and intercepts.
Exam Tips:
- Always state domain restrictions before solving.
- For exponentials, isolate before taking logs.
- For logs, ensure argument > 0 before proceeding.
- Practice switching between exponential and logarithmic forms quickly.