Exponential and Logarithmic Functions

Exponential functions have the form f(x) = aˣ where a > 0, a ≠ 1. The most important exponential function is eˣ (Euler's number, e ≈ 2.71828), which appears naturally in calculus due to its unique derivative property: d/dx(eˣ) = eˣ.

Logarithmic functions are the inverse of exponential functions. For y = aˣ, we write x = log_a(y). The natural logarithm ln(x) is the inverse of eˣ, meaning ln(eˣ) = x and e^(ln x) = x for x > 0.

Key Properties of Logarithms:

1. Product rule: log(AB) = log A + log B
2. Quotient rule: log(A/B) = log A − log B
3. Power rule: log(Aⁿ) = n log A
4. Change of base: log_a(x) = ln(x)/ln(a)
5. Special values: ln(1) = 0, ln(e) = 1

Important Equations:

• eˣ⁺ʸ = eˣ · eʸ
• (eˣ)ⁿ = eⁿˣ
• e⁰ = 1

Memory Aid (LOGS): L = Laws to Learn, O = One becomes zero (ln 1 = 0), G = Growth exponentially, S = Sum becomes product (ln(AB) = ln A + ln B)

The exponential graph y = eˣ passes through (0,1), increases rapidly, never touches the x-axis (horizontal asymptote y = 0). The logarithmic graph y = ln(x) is its reflection in y = x, passing through (1,0) with vertical asymptote x = 0, defined only for x > 0.

Exponential growth models countless real phenomena. Population growth follows P(t) = P₀e^(kt) where P₀ is initial population, k is growth rate, and t is time. If bacteria double every hour, they exhibit exponential growth—not linear. After 10 hours, you don't have 10× the bacteria; you have 2¹⁰ = 1024× more!

Compound interest demonstrates exponential functions: A = P(1 + r/n)^(nt) where P is principal, r is annual rate, n is compounds per year, and t is years. Continuously compounded interest uses A = Pe^(rt), showing why 'e' matters in finance.

Radioactive decay follows N(t) = N₀e^(-λt) where λ is the decay constant. Half-life occurs when N(t) = N₀/2, giving t_(1/2) = ln(2)/λ. Carbon-14 dating uses this principle with half-life 5,730 years._

pH scale in chemistry is logarithmic: pH = -log₁₀[H⁺]. Each unit decrease means 10× more acidic. pH 3 is 100× more acidic than pH 5.

Sound intensity (decibels) uses L = 10log₁₀(I/I₀). A whisper (30 dB) to normal conversation (60 dB) represents a 1000× increase in intensity, not double!

Analogy: Think of exponential growth like compound interest on knowledge—each piece you learn multiplies your understanding, not just adds to it. Logarithms are like asking "how many times must I multiply to reach this number?" If 2³ = 8, then log₂(8) asks "what power gives 8?" Answer: 3.

Example 1: Solve 3e^(2x-1) = 15 for x, giving your answer in exact form.

Solution:

1. Divide both sides by 3: e^(2x-1) = 5
2. Take natural log: ln(e^(2x-1)) = ln(5)
3. Use inverse property: 2x - 1 = ln(5)
4. Solve for x: 2x = 1 + ln(5)
5. x = (1 + ln 5)/2 ✓

Examiner note: Must show ln on both sides (1 mark), correct manipulation (1 mark), exact form required (1 mark).

Example 2: Solve log₂(x) + log₂(x-3) = 2

Solution:

1. Apply product rule: log₂[x(x-3)] = 2
2. Simplify: log₂(x² - 3x) = 2
3. Convert to exponential form: x² - 3x = 2²
4. Rearrange: x² - 3x - 4 = 0
5. Factorise: (x-4)(x+1) = 0
6. Solutions: x = 4 or x = -1
7. Reject x = -1 (logarithm undefined for negative)
8. Therefore x = 4 ✓

Examiner note: Must reject invalid solution with reasoning (1 mark).

Example 3: Differentiate y = ln(3x² + 5)

Solution: Using chain rule: dy/dx = 1/(3x² + 5) × 6x = 6x/(3x² + 5) ✓

Common error: Writing 1/(6x) instead of using chain rule properly.