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A2.4 · Solve applications using exponential and logarithmic graphs

Learn to solve applications using exponential and logarithmic graphs through clear examples and targeted practice.

Ontario Grade 12 Mathematics

Exponential and Logarithmic Functions

Use a model, a table, and a graph to answer questions about changing quantities.

A growing quantity may increase by the same percentage over equal time intervals. An exponential model can describe this pattern. A logarithmic graph can help answer a related question: how much time is needed to reach a chosen amount? In this lesson, you will connect a real situation to an equation, a table, and a graph. You will use the graph to estimate an answer and logarithms to calculate and check it.

What you will learn

1. Prerequisite bridge: read a model and a graph

A function is a rule that connects an input to an output. For an application, the input might be time and the output might be a population, balance, or amount of a substance. A graph shows how the output changes as the input changes.
An exponential function has a variable in the exponent. In a basic growth model, the starting amount is multiplied by the same factor for each equal time interval. The starting amount is the output when time is zero. A growth factor greater than one represents growth; a positive growth factor less than one represents decay.
A logarithm answers an exponent question. For example, asking for the time at which an exponential quantity reaches a target is asking what exponent produces that target. A logarithmic function is the inverse of an exponential function: it undoes the exponential relationship. This connection lets us solve for an unknown time.
Before using a model, identify what each variable measures and its unit. A solution of 8 is not useful by itself if the question asks for hours or years. Also consider which input values make sense. A model of elapsed time will usually use time values at or after the starting time.
y=a(b)ty=a(b)^t

2. From a situation to exponential and logarithmic graphs

In the model y=a(b)ty=a(b)^t, aa is the starting amount, bb is the growth or decay factor for one time interval, and tt is the number of time intervals. The output is yy. For example, if a quantity increases by 18% each hour, its hourly factor is 1.18 because the new amount is 100% plus 18% of the previous amount.
The exponential graph makes the pattern visible. For growth, it starts at aa when t=0t=0 and rises, often slowly at first and then more quickly. A table gives exact model outputs for selected inputs, while a graph shows the overall pattern between those inputs.
To find when the quantity reaches a target, draw or imagine a horizontal line at that target. Where it meets the exponential graph, read the corresponding input. This is a graphical estimate. A logarithmic equation can then give a more precise value for that input.
If the model has the form a(b)t=Ta(b)^t=T, where TT is the target, divide by aa and use a logarithm to isolate the exponent. The resulting time should agree with the graph estimate. When using a calculator, keep the same logarithm base in the numerator and denominator; common logarithms or natural logarithms both work.
A logarithmic graph can also show the same relationship with the roles of input and output reversed. In applications, it is often clearest to keep time on the horizontal axis and amount on the vertical axis, then use the logarithmic equation to calculate the time.
t=log⁡(T/a)log⁡bt=\frac{\log(T/a)}{\log b}

3. Guided application: estimate, calculate, and interpret

Suppose a culture begins with 240 cells and its population increases by 18% each hour. The model is P=240(1.18)tP=240(1.18)^t, where PP is the population after tt hours. We want to know when the model predicts 900 cells.
First make a rough prediction. After seven hours, the model gives about 764 cells. After eight hours, it gives about 902 cells. A graph of the model would meet the horizontal level P=900P=900 just before t=8t=8. The table narrows the answer to between seven and eight hours.
To calculate the crossing time, set the model equal to 900 and solve for the exponent. Dividing by 240 compares the target with the starting population. Taking logarithms then finds the exponent. This calculation gives about 7.99 hours, which is consistent with the graphical estimate.
The model describes a continuous curve, so its predicted value need not be a whole number of cells at every time. The question asks when the predicted amount reaches 900, so reporting approximately 7.99 hours is appropriate. If the question instead asked for the first whole-hour count at or above 900, the table shows that this occurs at eight hours.
900=240(1.18)t900=240(1.18)^t

4. A reliable approach and common errors

For an application, start by naming the input and output, including their units. Identify the initial amount and determine the factor for one time interval. Write the model, then decide whether the question asks for an amount or for a time. If it asks for an amount, substitute the time. If it asks for the time at a target, use the graph to estimate and a logarithm to calculate.
Check the result in two ways. First, substitute the calculated time into the model and see whether it gives a value close to the target. Second, compare the time with nearby table or graph values. A result outside the relevant part of the graph, or with the wrong unit, signals that something needs review.
A percentage is not itself the growth factor. An increase of 18% means multiplying by 1.18, not by 0.18. For a decrease of 18%, the factor would be 0.82. Also, the starting amount belongs at time zero; it is not an amount to add again after each interval.
When solving for time, do not stop after dividing the target by the initial amount. That ratio is the factor raised to the unknown time, not the time itself. A logarithm is needed to find the exponent. Finally, do not report more decimal places than the model or situation can support.
growth factor=1+r100\text{growth factor}=1+\frac{r}{100}

Selected values for the culture model

Time, t (hours)Population, P (cells)
0240
7about 764.4
7.99about 900
8about 902.0

Worked example

When will the culture reach 900 cells?

A culture starts with 240 cells and grows by 18% each hour. Use its exponential model to estimate when it reaches 900 cells. Give the time to the nearest hundredth of an hour.
  1. Build the model
    An increase of 18% means the amount is multiplied each hour by 1.18. The starting amount is 240, so use 240 as the value at time zero.
    P=240(1.18)tP=240(1.18)^t
  2. Use nearby values
    Evaluate the model at seven and eight hours. These values place the target between the two times and show where the graph crosses the horizontal level of 900 cells.
    P(7)≈764.4,P(8)≈902.0P(7)\approx764.4,\quad P(8)\approx902.0
  3. Solve for the crossing time
    Set the population equal to 900. Divide by the starting amount, then use logarithms to find the exponent. The graphical estimate is just before eight hours, so a result near eight hours is reasonable.
    t=log⁡(900/240)log⁡(1.18)≈7.99t=\frac{\log(900/240)}{\log(1.18)}\approx7.99
  4. Interpret the result
    The model predicts that the population reaches 900 cells at about 7.99 hours. This is a time, not a population count. At whole-hour readings, the first hour with a predicted amount at or above 900 is hour eight.
Answer: The model reaches 900 cells after approximately 7.99 hours.
Check: At seven hours the model predicts about 764 cells, below 900; at eight hours it predicts about 902 cells, just above 900. The calculated time lies between those values.

Common mistakes and how to avoid them

Using 0.18 as the growth factor for an 18% increase.
Correction: The new amount is 100% plus 18% of the old amount, so the factor is 1.18.
Treating the target divided by the initial amount as the time.
Correction: That ratio equals the growth factor raised to the time. Use a logarithm to find the exponent.
Reporting a numerical answer without its unit or context.
Correction: State what the answer measures, such as approximately 7.99 hours.
Assuming the graph estimate and calculator value must look exactly the same.
Correction: A graph gives an estimate based on its scale. Use the logarithmic calculation for precision, then check that it agrees with the graph.

Lesson summary

Check your understanding

Question 1

A quantity starts at 500 and increases by 6% each year. Which model represents the amount after t years?
  1. A=500(0.06)tA=500(0.06)^t
  2. A=500(1.06)tA=500(1.06)^t
  3. A=500+1.06tA=500+1.06t
  4. A=500(0.94)tA=500(0.94)^t
Show answer and explanation
A=500(1.06)tA=500(1.06)^t
An increase of 6% gives a multiplier of 1.06, applied once for each year.

Question 2

For the model A=500(1.06)tA=500(1.06)^t, what method finds the time when the amount reaches 800?
  1. Subtract 500 from 800 and call the result the time.
  2. Divide 800 by 500 and call the result the time.
  3. Set the model equal to 800 and use logarithms to find the exponent.
  4. Multiply 800 by 1.06.
Show answer and explanation
Set the model equal to 800 and use logarithms to find the exponent.
The unknown time is in the exponent. Setting the amount equal to 800 and using logarithms isolates that exponent.

Question 3

A graph crosses a target level between t=4 and t=5. What can you conclude from the graph?
  1. The target is reached between four and five time intervals.
  2. The target is reached at exactly four time intervals.
  3. The starting amount must be five.
  4. The growth factor must be less than one.
Show answer and explanation
The target is reached between four and five time intervals.
The intersection's horizontal position estimates the input at which the model reaches the target. A crossing between four and five means the time lies in that interval.

Key terms

Exponential model
A rule in which the variable appears in the exponent, often used when a quantity changes by the same factor over equal intervals.
Growth factor
The multiplier applied during each interval when a quantity grows.
Logarithm
A value that answers which exponent produces a specified result.
Target
The output value an application asks the model to reach.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation A2.4. It is a study resource, not an official curriculum publication.

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