5. Applications of Exponential and Logarithmic Functions
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Chapter 5
5.
Applications of Exponential and Logarithmic Functions
Exponential and logarithmic functions are not just abstract concepts; they have tangible applications in various fields. This lesson unveils the intricate ways these functions manifest in real-life situations. For instance, they play a crucial role in understanding phenomena like population growth, radioactive decay, and financial modeling. When we use exponential and logarithmic models, they offer a lens to predict and analyze these scenarios with greater accuracy. By embracing the practical implications of these mathematical tools, one gains a clearer perspective on the underlying mechanisms that drive numerous processes in our world.
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Lesson Settings & Tools
| | 9 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Applications of Exponential and Logarithmic Functions
Slide of 9
This lesson will work with real-life applications of exponential and logarithmic functions.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Graphing in the (x,ln y) Coordinate Plane
Exponential functions have nonlinear graphs. In the diagram, the graph of y=2^x and some of its points are shown.
Example
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Tables, Logarithmic Models, and Exponential Models
Heichi and Ignacio are good friends. However, when it comes to mathematics, they are pretty competitive. Neither Heichi nor Ignacio admits that the other person might be more knowledgeable in math.
Instead of letting this silly argument go, they decide to duel! They will challenge each other by asking one math question to the other.
a Ignacio presents a table that corresponds to the logarithmic function y=3-5 ln x.
| x | y |
|---|---|
| 1 | 3 |
| 2 | ≈ - 0.47 |
| 3 | ≈ - 2.49 |
| 4 | ≈ - 3.93 |
| 5 | ≈ - 5.05 |
{"type":"choice","form":{"alts":["Ignacio","Heichi"],"noSort":false},"formTextBefore":"","formTextAfter":"","answer":0}
b Heichi, waiting for this moment to outsmart Ignacio, pulls out the following table from his notebook.
| x | y |
|---|---|
| 1 | 1k |
| 2 | 2k |
| 3 | 4k |
| 4 | 8k |
| 5 | 16k |
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Solution
a To find out who is correct, the x-variable will be isolated. Start by using inverse operations to isolate the natural logarithm.
y=3-5ln x
▼
Solve for ln x
SubEqn
LHS-3=RHS-3
y-3=- 5ln x
DivEqn
.LHS /(- 5).=.RHS /(- 5).
y-3/- 5=ln x
RearrangeEqn
Rearrange equation
ln x=y-3/- 5
b Paying close attention to the y-values of the table, a constant multiplier can be identified.
It is seen that the constant multiplier is 2. Also, the initial value is 1k, or just k. With this information, an exponential function can be written. y= k( 2)^(x-1) Ignacio pats himself on the back.
Example
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Carbon 14 in Vegetables
Heichi is upset that he lost to Ignacio in the last round. He decides to take a breather and turn to his favorite science book to practice some other problems. Here is one of his favorite topics: It is estimated that a preserved vegetable contains about 1 microgram (a millionth of a gram) of Carbon 14.
The amount of Carbon 14 in the preserved vegetable can be modeled by the following exponential function. C(t)=I (1/2)^(t5500) Here, t is the time in years that has passed since the vegetable's death, and I is the amount of Carbon 14 present in the vegetable at death, measured in micrograms. Heichi, feeling good about seeing a familiar topic, dives into solving the following problems.
a If the vegetable died 2000 years ago, how much Carbon 14 was present in the living vegetable? Round the answer to two decimal places.
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b If the vegetable died 10 000 years ago, how much Carbon 14 was present in the living vegetable? Round the answer to two decimal places.
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c The amount it takes for half of the Carbon 14 to decay is called its half-life. What half-life does the expression for the given function C(t) imply for Carbon 14?
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Hint
a Substitute 2000 for t in the given function.
b Substitute 10 000 for t in the given function.
c The initial amount of Carbon 14 is I. Therefore, half the initial amount is I2.
Solution
a It is known that there is 1 microgram of Carbon 14 present in a vegetable that died 2000 years ago. To calculate how much Carbon 14 was present in the living vegetable, 2000 will be substituted for t and 1 will be substituted for C(2000). Then, the equation can be solved for I.
C(t)=I (1/2)^(t5500)
Substitute
t= 2000
C( 2000)=I (1/2)^(20005500)
Substitute
C(2000)= 1
1=I (1/2)^(20005500)
▼
Solve for I
ReduceFrac
a/b=.a /500./.b /500.
1=I (1/2)^(411)
PowQuot
(a/b)^m=a^m/b^m
1=I (1/2^(411))
MultEqn
LHS * 2^(411)=RHS* 2^(411)
2^(411)=I(1)
IdPropMult
Identity Property of Multiplication
2^(411)=I
RearrangeEqn
Rearrange equation
I=2^(411)
UseCalc
Use a calculator
I=1.286664...
RoundDec
Round to 2 decimal place(s)
I≈ 1.29
b Similar to Part A, it is known that there is 1 microgram of Carbon 14 present in a vegetable that died 10 000 years ago. This sentence can also be expressed mathematically.
C(10 000)=1
With this information, the amount of Carbon 14 that was present in the living vegetable I can be found by substituting 10 000 for t and 1 for C(10 000).C(t)=I (1/2)^(t5500)
Substitute
t= 10 000
C( 10 000)=I (1/2)^(10 0005500)
Substitute
C(10 000)= 1
1=I (1/2)^(10 0005500)
▼
Solve for I
ReduceFrac
a/b=.a /500./.b /500.
1=I (1/2)^(2011)
PowQuot
(a/b)^m=a^m/b^m
1=I (1/2^(2011))
MultEqn
LHS * 2^(2011)=RHS* 2^(2011)
2^(2011)=I(1)
IdPropMult
Identity Property of Multiplication
2^(2011)=I
RearrangeEqn
Rearrange equation
I = 2^(2011)
UseCalc
Use a calculator
I = 3.526365...
RoundDec
Round to 2 decimal place(s)
I≈ 3.53
c Half the initial amount of Carbon 14 present in the vegetable is I2. Therefore, the half-life of Carbon 14 present in the vegetable can be found by substituting I2 for C(t), and isolating the variable t.
C(t)=I (1/2)^(t5500)
Substitute
C(t)= I/2
I/2=I (1/2)^(t5500)
DivEqn
.LHS /I.=.RHS /I.
1/2=(1/2)^(t5500)
a=a^1
(1/2)^1=(1/2)^(t5500)
Example
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Uranium 238
Ignacio, feeling good about showing his prowess in math, goes to explore the chemistry lab. He sees none other than Heichi studying. Heichi invites Ignacio to see that Uranium 238 is the most common isotope of uranium found in nature. It also has many applications in nuclear technology. Uranium 238 decays exponentially and its half-life is about 4.5 billion years.
Ignacio feels a sense of joy that Heichi has shown him this.
a Heichi, unable to control his competitive urge, metaphorically flips the tables and once again challenges Ignacio. "Write an exponential function expressing how much Uranium 238 remains after t billion years if the initial sample size is 40 grams. If you can!"
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b Ignacio will not be taken down so easily. He asks Heichi which of the following graphs, in the context of this situation, corresponds to the function found in Part A.
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Hint
b Graph the function and compare it with the given choices. What is the y-intercept of the graph if the initial sample size is 40?
Solution
a Let U(t) be the amount of uranium 238 left from the sample, in grams, after t billions years. Knowing that the uranium decays exponentially, a general equation can be written.
U(t)=40e^(- kt)
Substitute
t= 4.5
U( 4.5)=40e^(- k( 4.5))
Substitute
U(4.5)= 20
20=40e^(- k(4.5))
DivEqn
.LHS /40.=.RHS /40.
20/40=e^(- k(4.5))
ReduceFrac
a/b=.a /20./.b /20.
1/2=e^(- k(4.5))
MultNegPos
(- a)b = - ab
1/2=e^(- 4.5k)
b It is now Heichi's turn to take on Ignacio's challenge to draw the graph of the function U(t)=40e^(- 0.154t). To do this, Heichi will make a table of values. Since time cannot be negative, only non-negative values of t will be considered.
| x | 40e^(- 0.154t) | y |
|---|---|---|
| 0 | 40e^(- 0.154( 0)) | 40 |
| 2 | 40e^(- 0.154( 2)) | ≈ 29.4 |
| 4 | 40e^(- 0.154( 4)) | ≈ 21.6 |
| 6 | 40e^(- 0.154( 6)) | ≈ 15.9 |
| 8 | 40e^(- 0.154( 8)) | ≈ 11.7 |
| 10 | 40e^(- 0.154( 10)) | ≈ 8.6 |
Example
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Modeling a Village's Population
Ignacio and Heichi see that this competition has turned ugly. They make a truce and decide to team up in studying for a geography test — together, as buddies.
However, the formulas did not allow the rivalry between Ignacio and Heichi to fade. While learning the concepts related to population, they find out that the population P of a small village in the north of Argentina is modeled by an exponential function. P(t)=500(1.04)^(2t) Here, t is the time in years that has passed since the foundation of the village in the year 2020. Heichi and Ignacio cannot help themselves and start challenging each other.
a Ignacio says that by the year 2050 the population, rounded to the nearest integer, will be about 5260 people. Is this true or false?
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b Heichi wants to find the year in which the population will be 3037 people, but does not know how to do it. Help Heichi find this year before Ignacio realizes that he cannot find the answer! Round the answer to the nearest integer.
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Hint
a How many years are there between 2020 and 2050?
b Substitute 3037 for P(t) and solve for t. Then, add this value to 2020.
Solution
a To determine whether Ignacio is correct, the population in the village for the year 2050 will be calculated. To do so, the difference between 2050 and 2020 needs to be found.
b To determine the year in which the population of the village will be about 3037 people, this number will be substituted for P(t). Then, the variable t will be isolated.
P(t)=500(1.04)^(2t)
Substitute
P(t)= 3037
3037=500(1.04)^(2t)
DivEqn
.LHS /500.=.RHS /500.
3037/500=(1.04)^(2t)
UseCalc
Use a calculator
6.074=(1.04)^(2t)
log_(1.04) 6.074 =2t
log_c a = log_b a/log_b c
log 6.074/log 1.04 =2t
▼
Solve for t
UseCalc
Use a calculator
45.996546... =2t
DivEqn
.LHS /2.=.RHS /2.
22.998273... =t
RearrangeEqn
Rearrange equation
t=22.998273...
RoundInt
Round to nearest integer
t≈ 23
Discussion
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Relation Between Exponential and Linear Models
Recall the exploration seen at the beginning. In general, a set of more than two points (x,y) fits an exponential model if and only if the set of transformed points (x,ln y) fits a linear model. For example, consider the following function. y=2^x This function can be graphed on an (x,y) coordinate plane.
y=2^x
ln(LHS)=ln(RHS)
ln y=ln 2^x
ln(a^b)= b*ln(a)
ln y=xln 2
| x | x ln 2 | ln y=xln 2 |
|---|---|---|
| - 2 | - 2ln 2 | ≈ - 1.39 |
| - 1 | - 1ln 2 | ≈ - 0.69 |
| 0 | 0ln 2 | 0 |
| 1 | 1ln 2 | ≈ 0.69 |
| 2 | 2ln 2 | ≈ 1.39 |
Example
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Visual Near Point
Heichi realizes that he might not be as great at math as Ignacio. To regain some confidence, he tells Ignacio that Ignacio's eyesight will worsen over time. "How can you prove that, Heichi?" asks Ignacio. "The visual near point of a person is the closest point at which an object can be placed and distinctly be observed by the person, and it varies with age," proudly states Heichi. 
Ignacio admits his interest in the topic. He decides to learn more and wants to create a scatter plot for the data pairs (x,ln y), where x is the age in years and y the distance of the visual near point in centimeters. Then, he needs to write an exponential model for the original data pairs (x,y). Give Ignacio a hand. 
Since the set of points (x,ln y) fits a linear model, an exponential model should be a good fit for the original data. Now recall the slope-intercept form of a line to write the equation for the line of fit. Remember that, in this case, the dependent variable is ln y, therefore the form of the equation should be as follows.
ln y=mx+b
Two points can be used to find the slope m. The points (20,ln 12) and (60,ln 100) can be substituted in the Slope Formula.
Knowing that the slope is about 0.053, a partial equation of the line can be written.
ln y=0.053x+b
To find the value of b, the intersection with the vertical axis, a point must be used. For simplicity, choose the first point of the table.
The equation of the line can now be written.
ln y=0.053x+1.42
Finally, to find the exponential model that represents the original data, the variable y must be isolated. To do so, the definition of a natural logarithm will be used.
ln y=0.053x+1.42 ⇕ y=e^(0.053x+1.42)
To simplify the obtained equation, the Product of Powers Property and the Power of a Power Property will be used.
Note that this is only an example exponential model. If any two other points are used, a different model will be obtained. Ignacio is happy to have solved this and has regained some trust in Heichi's math knowledge. Heichi, too, feels good about sharing a topic of interest with his friend.
Answer
Scatter Plot:
Example Exponential Model: y=4.137(1.054)^x
Hint
To create the scatter plot make a table of values and plot the obtained points on an (x,ln y) coordinate plane. To write the exponential model use any two of the obtained points.
Solution
To make the scatter plot, a table of data pairs (x,ln y) will be created using the given information.
| x | y | ln y |
|---|---|---|
| 20 | 12 | ln 12≈ 2.48 |
| 30 | 15 | ln 15≈ 2.71 |
| 40 | 25 | ln 25≈ 3.22 |
| 50 | 40 | ln 40≈ 3.69 |
| 60 | 100 | ln 100≈ 4.61 |
Next, the obtained points will be plotted on an (x,ln y) coordinate plane. Since all the values are positive, only the first quadrant will be considered.
The points are not collinear. However, they lie close to a straight line. Consider the line passing through the first and last points.
m=ln y_2-ln y_1/x_2-x_1
SubstitutePoints
Substitute ( 20,ln 12) & ( 60,ln 100)
m=ln 100- ln 12/60- 20
m≈ 0.053
y=e^(0.053x+1.42)
▼
Simplify right-hand side
SumInExponent
a^(m+n)=a^m*a^n
y=e^(0.053x)e^(1.42)
CommutativePropMult
Commutative Property of Multiplication
y=e^(1.42)e^(0.053x)
ProdInExponent
a^(m* n)=(a^m)^n
y=e^(1.42)(e^(0.053))^x
UseCalc
Use a calculator
y=4.137120...(1.054429...)^x
RoundDec
Round to 3 decimal place(s)
y=4.137(1.054)^x
Closure
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Using Technology
While it is extremely impressive to do math by hand, an exponential model for the last example can also be found by using a graphing calculator. Recall the given data points.
| x | y |
|---|---|
| 20 | 12 |
| 30 | 15 |
| 40 | 25 |
| 50 | 40 |
| 60 | 100 |
It was previously discussed that if graphing the data points in the form (x, ln y) results in a linear pattern, then an exponential function is a good model for the data points in the form (x,y). To find the equation of the exponential model using a graphing calculator, press STAT, choose EDIT,
and enter the values in the first two columns, L1 and L2.
Now, to find the values of ln y, place the cursor in the heading of L3 and press LN. After that, press 2ND, and then 2 before finally closing the parentheses. By pressing ENTER, L3 will be filled automatically with the corresponding values for ln y.
Having entered the values, they can be plotted in a scatter plot by pressing 2nd and Y=. Then, choose one of the plots in the list. Make sure to turn the desired plot ON. Choose the type to be a scatterplot, and assign L1 and L3 as XList and YList, respectively. Any mark can be picked.
After this, press GRAPH. If needed, the scale and the minimum and maximum x- and y-values can be adjusted by pressing WINDOW.
The points approximate a line. Therefore, an exponential function is an appropriate model. The best fitting exponential function for the given data can be found by performing an exponential regression. To do this press STAT, and from the CALC
menu choose the exponential regression option.
Here, a≈ 3.5 and b≈ 1.054, rounded to three decimal places. With this information, the exponential function can be written. y=3.5(1.054)^x
Note that the value of a is not the same as in the last example, but the value of b is exactly the same.
Applications of Exponential and Logarithmic Functions
Exercises
The table corresponds to an exponential function with the following form. y=ab^(x-1) Write the exponential function that corresponds to the table.
| x | y |
|---|---|
| 1 | 36 |
| 2 | 18 |
| 3 | 9 |
| 4 | 4.5 |
| 5 | 2.25 |
| 6 | 1.125 |
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Looking at the table, we see that the initial value is 36. We know that the table of values corresponds to an exponential function. Let's now identify the constant multiplier.
The constant multiplier is 12. We can use this and the initial value 36 to write our exponential function. y= 36( 1/2)^(x-1)
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The amount of Carbon-14 in micrograms in a preserved vegetable that died t years ago can be modeled by an exponential function. C(t)=1.29(1/2)^(t5500)
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We want to find the amount of Carbon-14 present in a vegetable that died 2000 years ago. To do so, we will substitute t=2000 in the equation for the exponential function. Then, we will evaluate the right-hand side. Let's do it!
We found that a preserved vegetable that died 2000 years ago would contain about 1 microgram of Carbon-14.
A preserved vegetable contains 1.25 micrograms of Carbon-14. We want to find how many years ago it died. To do so, we will substitute 1.25 for C(t) in the given formula and solve for t.
Now we will use the definition of a logarithm to rewrite the exponential equation as a logarithmic equation. Definition b^c= a ⇔ c=log_b a [1em] Equation ( 1/2)^(t5500)= 125/129 ⇔ t/5500=log_(12) 125/129 We will now solve the logarithmic equation for t. To do this, we will use the Change of Base Formula so that we can use a calculator to find the numeric value of the logarithm on the right-hand side. Recall that log a is equivalent to log_(10) a.
The vegetable died about 250 years ago.
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During the summer, the number of surfboards sold in a Hawaiian surf shop grows exponentially. The owner knows that the number of boards sold during this season can be modeled by an exponential function.
B(t)=a(b)^(t-1)
Here, B(t) represents the number of boards sold and t is the number of days that have passed since the season started. On the first day of the season, 2 boards where sold. In the tenth day, 5 boards were sold. Find the values of a and b and write the exponential function. If necessary, round to five decimal places.
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We want to find the values of a and b and write the equation of the exponential function that models the given situation. Let's start by finding a. We know that 2 boards were sold on the first day of the season, so we can substitute t=1 and B(t)=2 in the given equation and solve for a.
We found that a=2. Let's put it into our equation. B(t)=2(b)^(t-1) We also know that 5 boards were sold on the tenth day of the season. This means that we can substitute B(t)=5 and t=10 into our partial equation and solve for b.
Now that we have the value of b, we can write the complete exponential function. B(t)=2(1.10717)^(t-1)
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During the summer, the number of surfboards sold in a Hawaiian surf shop grows exponentially. The owner knows that the number of boards sold during this season can be modeled by an exponential function. B(t)=2(1.10717)^(t-1) Here, B(t) represents the number of boards sold and t is the number of days that have passed since the season started. Select the graph that corresponds to the exponential function.
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To determine which of the graphs corresponds with the given exponential function, we will draw the graph of the function and compare it to the diagrams in the options. Let's start by make a table of values. Since the domain of an exponential function is the set of all real numbers, we can assign any value to the variable t.
| t | 2(1.10717)^(t-1) | B(t) |
|---|---|---|
| - 6 | 2(1.10717)^(- 6-1) | ≈ 1 |
| - 4 | 2(1.10717)^(- 4-1) | ≈ 1.2 |
| - 2 | 2(1.10717)^(- 2-1) | ≈ 1.5 |
| 0 | 2(1.10717)^(0-1) | ≈ 1.8 |
| 2 | 2(1.10717)^(2-1) | ≈ 2.2 |
| 4 | 2(1.10717)^(4-1) | ≈ 2.7 |
| 6 | 2(1.10717)^(6-1) | ≈ 3.3 |
We can now plot the points obtained in the table and connect them with a smooth curve.
Our graph matches the one given in option A.
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The Spanish government is opening factories in small towns to encourage people to move there. From the moment a massive factory was opened, the population of a small town started to increase year after year. The population P of this town, in thousands of people, can be modeled by an exponential function.
P(t)=10(1.02)^t
Here, t represents the number of years that have passed since the factory was opened. How many people will there be in the town 50 years after the factory's opening? Round the answer to the nearest thousand.
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We want to find how many people there will be in the town 50 years after the factory was opened. To do so, we will substitute t=50 in the given equation for the exponential function. Then, we will evaluate the right-hand side. Let's do it!
The population will be 26.915880... thousand people. Therefore, rounded to the nearest thousand, the population will be about 27 000 people.
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A store sells skateboards. The table shows the numbers of skateboards sold s during the nth month that the store has been open.
| n | s |
|---|---|
| 1 | 12 |
| 2 | 16 |
| 3 | 25 |
| 4 | 36 |
| 5 | 50 |
Which of the following is a scatter plot on the (n,ln s) plane of the data shown in the table?
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Let's start by calculating the natural logarithm of the s-values in the table.
| n | s | ln s |
|---|---|---|
| 1 | 12 | ln 12≈ 2.5 |
| 2 | 16 | ln 16≈ 2.8 |
| 3 | 25 | ln 25≈ 3.2 |
| 4 | 36 | ln 36≈ 3.6 |
| 5 | 50 | ln 50≈ 3.9 |
Now, let's plot the points (n,ln s) on a coordinate plane were the horizontal axis represents the variable n and the vertical axis represents the natural logarithm of s.
This graph corresponds to choice C.
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Applications of Exponential and Logarithmic Functions
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ln
log▢
1
2
3
()
∣
sin
cos
tan
0
.
π
x
y
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