Exponents/logarithms in modelling

Exponential functions model situations where quantities grow or decay at rates proportional to their current value, expressed as f(x) = a·b^x where a is the initial value, b is the base (growth factor), and x is the independent variable (often time). When b > 1, we have exponential growth; when 0 < b < 1, we have exponential decay.

Logarithms are the inverse operations of exponents. If b^y = x, then log_b(x) = y. The natural logarithm uses base e (≈2.718): ln(x) = log_e(x). Key logarithm laws include:

• Product Rule: log_b(xy) = log_b(x) + log_b(y)
• Quotient Rule: log_b(x/y) = log_b(x) - log_b(y)
• Power Rule: log_b(x^n) = n·log_b(x)
• Change of Base: log_b(x) = ln(x)/ln(b)

Critical IB Connection: Linearization transforms exponential models into linear form. Taking logarithms of both sides of y = ab^x gives ln(y) = ln(a) + x·ln(b), creating a linear relationship between ln(y) and x.

The exponential model P(t) = P₀e^(kt) is fundamental, where P₀ is the initial quantity, k is the growth/decay constant (positive for growth, negative for decay), and t is time. The doubling time satisfies 2P₀ = P₀e^(kt_d), giving t_d = ln(2)/k. Similarly, half-life t_h = ln(2)/|k| for decay models.

Command words in IB include: model (create mathematical representation), determine (find specific values), interpret (explain meaning in context), and justify (provide mathematical reasoning).

Population Growth: Bacterial colonies, human populations (under ideal conditions), and viral spread follow exponential models. Consider a bacteria population doubling every 3 hours. Starting with 100 bacteria, after t hours: P(t) = 100 × 2^(t/3). This models unrestricted growth until resources become limited.

Radioactive Decay: Carbon-14 dating uses exponential decay with a half-life of 5,730 years. If a fossil contains 25% of original Carbon-14, we solve 0.25 = e^(-kt) where k = ln(2)/5730. This gives t ≈ 11,460 years.

Financial Mathematics: Compound interest demonstrates exponential growth. An investment of £5,000 at 4% annual interest compounds to A = 5000(1.04)^t. After 10 years: A = £7,401.22. The continuous compounding formula A = Pe^(rt) models scenarios where interest compounds infinitely often.

Medicine: Drug concentration in bloodstream follows exponential decay. If a 100mg dose has a half-life of 6 hours, the model C(t) = 100e^(-kt) where k = ln(2)/6 ≈ 0.1155 predicts concentration at any time.

Think of exponential growth like a snowball rolling downhill — it starts small but accelerates rapidly as it gains mass. Each rotation adds proportionally more snow to its surface.

pH Scale and Logarithms: pH = -log₁₀[H⁺] converts hydrogen ion concentration to a manageable scale. A pH difference of 1 represents a 10-fold change in acidity. Lemon juice (pH 2) is 100 times more acidic than tomatoes (pH 4).

Sound Intensity: The decibel scale uses logarithms: L = 10log₁₀(I/I₀) where I₀ is the reference intensity. This compresses the enormous range of human hearing into a practical 0-140 scale.

Example 1: A radioactive substance has a half-life of 8 days. If 200g remains after 20 days, determine the initial mass.

Solution: Use M(t) = M₀e^(-kt) where k = ln(2)/8 = 0.08664.

Substitute known values: 200 = M₀e^(-0.08664×20)

200 = M₀e^(-1.7328) → 200 = M₀(0.1768) → M₀ = 1,131g

Examiner Note: Always show the substitution step clearly. 2 marks for correct model, 1 mark for k-value, 2 marks for final answer.

Example 2: The population of a town grows according to P(t) = 45000(1.025)^t. When will the population reach 60,000?

Solution: Set up equation: 60000 = 45000(1.025)^t

Divide both sides: 1.333... = (1.025)^t

Take logarithms: ln(1.333) = t·ln(1.025)

0.2877 = t(0.02469) → t = 11.65 years

IB Requirement: Round appropriately and interpret in context: "The population will reach 60,000 during the 12th year."

Example 3: Linearize y = 5(3^x) and find the gradient.

Solution: Take natural logarithm: ln(y) = ln(5) + x·ln(3)

This is linear form Y = c + mx where Y = ln(y), c = ln(5) = 1.609, m = ln(3) = 1.099

Gradient = 1.099

Key Skill: Recognizing that ln(3) appears as the coefficient of x earns the method mark even if calculations contain minor errors.