Year 12 Exponentials and logarithms: Growth and decay

Exponential models are used when a quantity changes by the same percentage or scale factor over equal time intervals.

Growth uses a factor greater than 1, such as

1.04

. Decay uses a factor between 0 and 1, such as

0.92

Main models

Growth: $A = A_{0} {(1 + r)}^{t}$
Decay: $A = A_{0} {(1 - r)}^{t}$

Worked example

Problem: A town has a population of 18,000 and grows by 4% per year. Predict the population after 6 years.

Initial population: $A_{0} = 18000$ .
Growth rate: $r = 0.04$ .
Time: $t = 6$ .
Substitute into the model: $A = 18000 {(1.04)}^{6}$ .
Evaluate the power: ${(1.04)}^{6} = 1.265319 . . .$ .
Multiply: $A = 18000 \times 1.265319 . . . = 22775.742 . . .$ .
Round to the nearest person: $A = 22776$ .

Answer: the predicted population after 6 years is about

22, 776

Reading the growth factor correctly

Many students lose marks before they even begin the model because they use the percentage as the factor. A growth rate of 4 percent does not mean multiply by $0.04$ . It means keep the original amount and add 4 percent more, so the factor is $1.04$ . Similarly, a decay rate of 8 percent means keep 92 percent, so the factor is $0.92$ .

Worked example 2

Problem: A machine worth

15000

dollars depreciates by 12 percent each year. Find its value after 5 years.

Initial value: $A_{0} = 15000$ .
Decay rate: $r = 0.12$ , so the decay factor is $0.88$ .
Model: $A = 15000 {(0.88)}^{5}$ .
Evaluate the power: ${(0.88)}^{5} = 0.527731 . . .$ .
Multiply: $A = 15000 x 0.527731 . . . = 7915.97 . . .$ .
Round appropriately: $A = 7916$ dollars, to the nearest dollar.

Answer: the machine is worth about

7916

dollars after 5 years.

Common mistakes

Using $0.12$ instead of $0.88$ for 12 percent decay.
Mixing up the time unit in the question and the time unit in the rate.
Rounding the growth factor too early.
Forgetting to interpret the final answer in context.

Revision focus

Exponential and logarithmic questions become much easier when you classify the task before starting. Ask whether you are evaluating a logarithm exactly, simplifying with log laws, solving an exponential equation, solving a logarithmic equation, or modelling growth and decay. Each of those tasks has a standard pattern, and confusion usually comes from mixing those patterns together rather than from the algebra itself.

Another high-value habit is to reverse-check the answer. If a logarithm equals a certain number, convert it back into exponential form and see whether the statement is true. If a model gives a time or population value, ask whether the rounding and units make sense in context. Domain restrictions, positive log arguments, and sensible growth factors are not side details in this topic. They are part of what makes the answer mathematically valid.

A quick final check for this topic is to ask whether the form of the answer matches the form of the question. Exact evaluations should usually stay exact. Solving questions should respect domain restrictions. Modelling questions should include sensible units, rounding, and interpretation. Those checks are short, but they are exactly the checks that stop a nearly correct solution from becoming an incorrect one.