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Master exponential and logarithmic functions for AP Precalculus with a 1-hour mega test, detailed answer key, interactive mind map, revision notes, and modeling questions. Covers sequences, growth & decay, logarithmic laws, composition, inverses, semi-log plots, and more. Perfect for exam preparation.
Mind Map — Exponential & Logarithmic Functions
1. Arithmetic vs Geometric Change
• Arithmetic → constant difference → an = a1 + (n − 1)d
• Geometric → constant ratio → an = a1 rn − 1
• Identify: difference → linear, ratio → exponential
2. Linear vs Exponential Models
• Linear: y = mx + b → constant rate
• Exponential: y = a · bx → percentage rate
• b > 1 → growth, 0 < b < 1 → decay
3. Logs & Exponents (Inverse Relationship)
• logb(a) = x ↔ bx = a
• ln x = log base e
• eln x = x and ln(ex) = x
4. Log Laws
• log(ab) = log a + log b
• log(a / b) = log a − log b
• log(an) = n · log a
5. Exponential Models
• y = a ekt
• k > 0 → growth, k < 0 → decay
• Half-life → T1/2 = (ln 2) / |k|
6. Inverse Functions
• Swap x and y, then solve for y
• Example: y = 5 e2x → ln(y / 5) = 2x → x = (1/2) ln(y / 5)
7. Composition of Functions
• (f ∘ g)(x) = f(g(x))
• Substitute g(x) completely into f(x)
• Example: if f(x) = 2x, g(x) = ln x → (f ∘ g)(x) = 2ln x
8. Solving Exponential & Log Equations
• Exponential: ax = b → x = loga(b)
• Logarithmic: loga(x) = k → x = ak
• Use log laws to simplify first
9. Semi-Log Plots
• y-axis logarithmic + x-axis linear
• Exponential function → straight line when log(y) plotted vs x
• Line form: log(y) = m x + c
10. Model Validation — RSS
• RSS = Σ (Actual − Predicted)2
• Lower RSS → better model