Year 12 Exponentials and logarithms: Word problems

In many applications the unknown is the time, so the variable sits in the exponent. That is where logarithms become the tool that unlocks the model.

First build the exponential equation, then isolate the power, then take logs on both sides.

Main method

Write the exponential model from the context.
Substitute the known values.
Isolate the exponential expression.
Take $\log$ or $\ln$ of both sides and solve for the variable.
Interpret the answer in context, including any rounding.

Worked example

Problem: An investment grows according to

V = 8000 {(1.06)}^{t}

. After how many years will it first exceed

12000

dollars?

Set the model equal to 12000: $12000 = 8000 {(1.06)}^{t}$ .
Divide by 8000: $1.5 = {(1.06)}^{t}$ .
Take natural logs: $\ln (1.5) = \ln ({(1.06)}^{t})$ .
Use the power rule: $\ln (1.5) = t \ln (1.06)$ .
Solve for $t$ : $t = \ln (1.5) / \ln (1.06)$ .
Evaluate: $t = 6.96 . . .$ .
The investment first exceeds 12000 after the next whole year, so $t = 7$ years.

Answer: the investment first exceeds

12000

dollars after

7

years.

How to read the wording

Most modelling mistakes happen before the logarithms appear. Students often use the wrong initial value, the wrong growth factor, or the wrong time unit. A clean setup should identify what is changing, the starting amount, the percentage rate, the period for that rate, and the quantity the question is asking for. Once that model is correct, the logarithmic solving stage is mostly routine.

Worked example 2

Problem: A culture starts with 600 bacteria and doubles every 3 hours. How long will it take for the culture to reach 5400 bacteria?

Use the doubling model: $N = 600 x 2^{t / 3}$ .
Set the population equal to 5400: $5400 = 600 x 2^{t / 3}$ .
Divide by 600: $9 = 2^{t / 3}$ .
Take logs: $\log 9 = (t / 3) \log 2$ .
Solve for $t$ : $t = 3 \log 9 / \log 2$ .
Evaluate: $t = 9.51 . . .$ hours.

Answer: it takes about

9.5

hours for the culture to reach 5400 bacteria.

Common mistakes

Taking logs before isolating the exponential term.
Using years in the exponent when the rate is given per month, or vice versa.
Forgetting that "first exceeds" may require rounding up to the next full time unit.
Giving a numerical answer without units.

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.