Modelling with Exponentials & Logarithms

Chapter Overview

• Exponential Growth & Decay: Functions of the form y = Ae^(kt) for growth and y = Ae^(-kt) for decay, where the rate of change is always proportional to the current value
• Initial Values & Constants: A is always the starting value when t = 0; k > 0 always, with the sign in the exponent determining growth or decay
• Real-World Modelling: Population growth, radioactive decay, drug elimination, compound interest, and depreciation all follow exponential patterns
• Converting Bases: Any exponential function a^x can be rewritten in the form e^(kx) using the fact that a^x = e^(ln(a)·x)
• E and ln as Inverses: ln is the natural logarithm (inverse of e); ln(e^x) = x and e^(ln(x)) = x always hold
• Logarithmic Axes & Linearisation: Taking logs of exponential equations transforms curved graphs into straight lines, making it easy to extract unknown constants
• Three Key Transformations: Type 1 (y = Ab^x), Type 2 (y = Ae^(kx), special case), Type 3 (y = Ax^b), each producing a different linearised form
• Model Limitations: Real-world exponential models only apply within certain time ranges; populations have limits, old objects lose all value, radioactivity levels off

1. Exponential Growth & Decay

What Does "Exponential" Mean?

Exponential growth and decay describe situations where the rate of change is proportional to how much you already have. Think of a bank account earning compound interest: the more money you have, the more interest you earn. Or a viral video: the more people who've seen it, the faster it spreads.

The key defining feature is that the exponent (the power) contains the variable, not just a coefficient. This makes the growth or decay accelerate over time.

Exponential Growth: y = Ae^(kt)

For exponential growth, we use:

What each letter means:

• A = the starting or initial value (when t = 0, y always equals A)
• e ≈ 2.71828... (Euler's number — a special mathematical constant)
• k = the growth constant (always positive; larger k means faster growth)
• t = time (we use t instead of x because time-dependent models are common)

The graph starts at point (0, A) and curves upward, accelerating as t increases. It increases slowly at first, then rapidly.

Exponential Decay: y = Ae^(−kt)

For exponential decay, the only difference is a minus sign in the exponent:

The graph starts at (0, A) and curves downward, decreasing rapidly at first, then more slowly, approaching zero but never reaching it. This models radioactive decay, drug levels in blood, or the value of a depreciating car.

Converting Any Base to e Form

2. Using Exponentials in Modelling

What Is Exponential Modelling?

3. Transforming Graphs Using Logarithms