Learning Objectives
Build an exponential model from a constant percentage change.
Interpret a logarithm as an exponent and state its domain conditions.
Solve for the time at which an exponential model reaches a target.
Prerequisites: geometric sequences and exponent rules. This is a mathematical modeling lesson; the numerical contexts are illustrative. Check whether a process is measured continuously or only at whole-number stages.
1. Use a Multiplier for Repeated Percentage Change
An exponential model has the form y = Abᵗ, where A is the value at t = 0 and b > 0 is the multiplier for one time unit. Growth by p% uses b = 1 + p/100. Decay by p%, for 0 < p < 100, uses b = 1 − p/100.
Worked Example: Repeated Decay
A sample starts at 500 grams and retains 80% of its mass after each modeled day. Its mass is M(t) = 500(0.8)ᵗ grams.
After three days, M(3) = 500(0.512) = 256 grams.
The daily loss is not a fixed 100 grams. The first loss is 100 grams, but the second is 20% of 400 grams, which is 80 grams.
2. A Logarithm Finds an Exponent
For b > 0, b ≠ 1, and x > 0, logᵦ(x) = y means bʸ = x. For example, log₂(8) = 3 because 2³ = 8. Natural logarithms use base e and are written ln. Logarithms allow us to isolate a variable that appears in an exponent.
Law | Conditions |
|---|---|
ln(xy) = ln x + ln y | x > 0 and y > 0 |
ln(x/y) = ln x − ln y | x > 0 and y > 0 |
ln(xᵏ) = k ln x | x > 0; k real |
These are product, quotient, and power laws. There is no matching rule that splits ln(x + y) into ln x + ln y. For example, ln(1 + 1) = ln 2, whereas ln 1 + ln 1 = 0.
3. Solve for a Target Time
Worked Example: Reaching a Growth Target
A model is N(t) = 200(1.5)ᵗ. Find when N(t) = 1000.
Divide by 200 to obtain (1.5)ᵗ = 5. Take natural logarithms: t ln(1.5) = ln 5.
Therefore t = ln 5 / ln(1.5) ≈ 3.97 time units. Keep the exact logarithmic expression until the final rounding.
If measurements occur only after whole stages, the first stage at or above 1000 is stage 4. Check: N(3) = 675 and N(4) = 1012.5.
For a decaying model, ln b is negative. The calculation is still valid, but inequality manipulations must respect the sign. A reliable interpretation check is to evaluate the model at the neighboring whole-number times before making a discrete conclusion.
4. State What the Model Assumes
An exponential model assumes the multiplier stays constant over equal time intervals. Real data may only approximately follow this pattern. Report the units of t and y, the starting value, and the range over which the model is intended to apply. Predictions far beyond observed data need a separate justification.
Common Mistakes
A 6% increase means multiply by 1.06, not 0.06 or 6.
Check that logarithm arguments are positive in real-number calculations.
A fractional target time and the first whole stage reaching a target answer different questions.
5. Practice
1. Write a model for an initial quantity of 80 that decreases by 25% each hour. Find its value after two hours.
2. Solve 3ˣ = 20, giving an exact expression.
3. A sequence starts at 10 and doubles each stage. How many completed stages are first needed to reach at least 100?
Worked Solutions
1. Q(t) = 80(0.75)ᵗ. At t = 2, Q(2) = 45.
2. Taking logarithms gives x ln 3 = ln 20, so x = ln 20 / ln 3.
3. After three stages the quantity is 80; after four it is 160. The first qualifying stage is 4. The initial quantity is counted as stage 0.
6. Summary
- Repeated percentage change uses a multiplier raised to the number of intervals.
- Logarithms invert exponentiation under their domain conditions.
- Interpret target times according to the model's continuous or discrete setting.