Exponential Modeling

The initial value a is the value of the quantity at time t = 0 (or at whatever starting point you choose). In a word problem, look for phrases like 'initially,' 'at the start,' or 'at time zero.' For example, if a bacterial culture starts with 500 cells, then a = 500. Mathematically, a = f(0) because b⁰ = 1, so f(0) = a · b⁰ = a.

The base b determines whether the model represents growth (b > 1) or decay (0 < b < 1). If the quantity grows by r% each period, then b = 1 + r/100. If it decays by r%, then b = 1 − r/100. For example, a 6% annual growth rate gives b = 1.06, while a 15% annual decay rate gives b = 0.85. If you are given two data points (t₁, y₁) and (t₂, y₂), you can solve for b algebraically: divide the two equations to get y₂/y₁ = b^(t₂ − t₁), then isolate b = (y₂/y₁)^(1/(t₂ − t₁)).

Combine a and b into the standard form f(t) = a · bᵗ, where t represents time (or whatever independent variable is appropriate). Always specify the units of t — years, hours, days, etc. — so the model is unambiguous. If the context provides a continuous rate k instead of a per-period rate, use the continuous form f(t) = a · eᵏᵗ, where e ≈ 2.71828 is Euler's number. The two forms are connected by b = eᵏ, or equivalently k = ln(b).

Once the model is written, substitute specific values of t to make predictions (interpolation within the data range, or extrapolation beyond it). You can also solve for t when a particular output value is given — this requires logarithms. For instance, to find when f(t) = C, write a · bᵗ = C, divide both sides by a, then take the logarithm: t = log(C/a) / log(b). Always check that your answer is reasonable given the real-world context.

Before applying an exponential model, verify that the data actually behaves exponentially. In a table of evenly spaced inputs, check successive ratios y₂/y₁, y₃/y₂, etc. If these ratios are approximately constant, an exponential model is appropriate. If the differences y₂ − y₁, y₃ − y₂ are approximately constant instead, a linear model is better. This diagnostic check is essential to avoid misapplying the wrong family of functions.

• f(t) — Output value (depends on context): The quantity being modeled at time t (e.g., population, amount of substance, account balance).
• a — Initial value (same as f(t)): The value of the quantity when t = 0; must be positive for a standard exponential model.
• b — Base / growth or decay factor (dimensionless): The multiplicative factor applied each unit of time. b > 1 gives growth; 0 < b < 1 gives decay.
• t — Time (or other independent variable) (years, hours, etc.): The exponent; represents how many time periods have elapsed since the start.

Problem. A town had a population of 12,000 in 2010. By 2018 the population had grown to 17,294. (a) Find an exponential model for the population. (b) Predict the population in 2025. (c) In what year will the population reach 25,000?

Answer. Model: P(t) = 12,000 · (1.0467)ᵗ. Predicted 2025 population ≈ 23,771. Population reaches 25,000 around the year 2026.

• Introduction to Exponential Functions — Exponential modeling builds directly on the definition and properties of exponential functions — you must understand the base form f(x) = abˣ before constructing models from data.
• Introduction to Logarithms — Solving for the time variable in an exponential model (e.g., finding when a population doubles) requires inverting the exponential using logarithms.
• Half-Life and Doubling Time — Half-life (decay) and doubling time (growth) are special benchmark predictions derived directly from exponential models and are among their most common applications.
• Continuous Exponential Growth (f = aeᵏᵗ) — The continuous form of the exponential model uses Euler's number e and the continuous growth rate k, which arises naturally in compound interest and population ecology.