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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.

Explore

Graphing in the Coordinate Plane

Exponential functions have nonlinear graphs. In the diagram, the graph of and some of its points are shown.

The graph of the exponential function and some points on it
Consider transforming these points into a different coordinate plane, where the horizontal axis represents the variable and the vertical axis represents To do so, manipulate to find a rule for then make a table of values for values of between and Finally, locate the points in the following coordinate plane to draw the graph.
line on the (x,ln y) coordinate plane
Identify the shape of the curve. When any exponential function is drawn on this coordinate plane, does a similar curve occur?
Example

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
Ignacio says it is impossible to find a logarithmic model of the form for this table. Heichi quickly counters, "It is totally possible." Ignacio replies, "Huh? That is bananas. No way, Heichi." Who is correct?
b Heichi, waiting for this moment to outsmart Ignacio, pulls out the following table from his notebook.
Assuming that is a positive number, Heichi challenges Ignacio to write as an exponential function of in its simplest form. Help Ignacio solve Heichi's challenge!

Hint

a Isolate the variable What type of function is the resulting function?
b What is the constant multiplier between consecutive values?

Solution

a To find out who is correct, the variable will be isolated. Start by using inverse operations to isolate the natural logarithm.
Solve for
Now, use the definition of a natural logarithm to isolate
When is the independent variable and the dependent variable, an exponential function is obtained.
Therefore, it is not possible to find a logarithmic model when is the independent variable and the dependent variable. This means that a logarithmic model of the form is not possible for this table. Ignacio is correct.
b Paying close attention to the values of the table, a constant multiplier can be identified.
constant multiplier
It is seen that the constant multiplier is Also, the initial value is or just With this information, an exponential function can be written.
Ignacio pats himself on the back.
Example

Carbon 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 microgram (a millionth of a gram) of Carbon

PlantFossil.jpeg

The amount of Carbon in the preserved vegetable can be modeled by the following exponential function.
Here, is the time in years that has passed since the vegetable's death, and is the amount of Carbon 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 years ago, how much Carbon was present in the living vegetable? Round the answer to two decimal places.
b If the vegetable died years ago, how much Carbon was present in the living vegetable? Round the answer to two decimal places.
c The amount it takes for half of the Carbon to decay is called its half-life. What half-life does the expression for the given function imply for Carbon

Hint

a Substitute for in the given function.
b Substitute for in the given function.
c The initial amount of Carbon is Therefore, half the initial amount is

Solution

a It is known that there is microgram of Carbon present in a vegetable that died years ago. To calculate how much Carbon was present in the living vegetable, will be substituted for and will be substituted for Then, the equation can be solved for
Solve for
Therefore, about micrograms of Carbon were present in the living vegetable.
b Similar to Part A, it is known that there is microgram of Carbon present in a vegetable that died years ago. This sentence can also be expressed mathematically.
With this information, the amount of Carbon that was present in the living vegetable can be found by substituting for and for
Solve for
Therefore, about micrograms of Carbon were present in the living vegetable.
c Half the initial amount of Carbon present in the vegetable is Therefore, the half-life of Carbon present in the vegetable can be found by substituting for and isolating the variable

Next, the Property of Equality for Exponential Equations can be used.
Finally, the value of can be found by multiplying both sides of the obtained equation by
Therefore, the half-life of Carbon implied by the given function is years. Heichi is feeling re-energized after solving for these solutions.
Example

Uranium

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 is the most common isotope of uranium found in nature. It also has many applications in nuclear technology. Uranium decays exponentially and its half-life is about billion years.

Uranium238.jpeg

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 remains after billion years if the initial sample size is grams. If you can!"
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.
graphs
Help Heichi determine the graph!

Hint

a Knowing that the uranium decays exponentially, the general equation for the remaining amount of uranium after billion years is where and are positive numbers.
b Graph the function and compare it with the given choices. What is the intercept of the graph if the initial sample size is

Solution

a Let be the amount of uranium left from the sample, in grams, after billions years. Knowing that the uranium decays exponentially, a general equation can be written.
Here, and are positive real numbers, where is the initial amount. Since the initial sample size is grams,
Furthermore, the half-life of uranium is billion years. Therefore, is equal to With this information, starting with substitution, the value of can be found.
Now the definition of a natural logarithm can be applied to remove the variable from the exponent.
Finally, the value of can be found by dividing both sides of the obtained equation by
Now that the value of was found, the desired function can be written.
Ignacio, once again, feels he has defeated Heichi's challenge. But wait! Heichi points out there is a mistake in Ignacio's calculation. The value of is approximately as found before, and not as Ignacio wrote in the function rule. This means that the following is the desired function.
Ignacio gasps in despair, as Heichi gloats.
b It is now Heichi's turn to take on Ignacio's challenge to draw the graph of the function To do this, Heichi will make a table of values. Since time cannot be negative, only non-negative values of will be considered.
Now, the points can be plotted and connected with a smooth curve. Recall that cannot take negative values. Therefore, the negative part of the horizontal axis will not be considered.
graph of exponential funtion
This graph corresponds to choice A. Heichi answered correctly. He shrugs his shoulders and is ready to crown himself as the champion of Math.
Example

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.

WorldGamze.png

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 of a small village in the north of Argentina is modeled by an exponential function.
Here, is the time in years that has passed since the foundation of the village in the year Heichi and Ignacio cannot help themselves and start challenging each other.
a Ignacio says that by the year the population, rounded to the nearest integer, will be about people. Is this true or false?
b Heichi wants to find the year in which the population will be 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.

Hint

a How many years are there between and
b Substitute for and solve for Then, add this value to

Solution

a To determine whether Ignacio is correct, the population in the village for the year will be calculated. To do so, the difference between and needs to be found.
Next, will be substituted into the given equation.
Evaluate right-hand side
The population of the village by the year will be about people. Therefore, Ignacio is correct. His eyes are gleaming with pride.
b To determine the year in which the population of the village will be about people, this number will be substituted for Then, the variable will be isolated.
Now, to eliminate the from the exponent, the definition of a logarithm can be used.
Next, the Change of Base Formula will be used to find the value of

Solve for
The population of the village will be people about years after its foundation. This is in the year That was super close — Ignacio almost noticed Heichi's struggle but not before Heichi found the solution.
Discussion

Relation Between Exponential and Linear Models

Recall the exploration seen at the beginning. In general, a set of more than two points fits an exponential model if and only if the set of transformed points fits a linear model. For example, consider the following function.
This function can be graphed on an coordinate plane.
exponential curve
Now, considering the given function, natural logarithms can be taken on both sides of the equation. Then, to simplify, the Power Property of Logarithms can be used.

Next, will be graphed on an coordinate plane. To do this, a table of values will be constructed.
Now, the points found in the table can be plotted and connected on an coordinate plane.
ln y = x ln 2
While the graph of is an exponential curve in the plane, the graph of is a straight line in the plane. It is worth noting that, in the plane, the equation of a line is where is the slope and the intercept.
Example

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.
visual near point distances
Ignacio admits his interest in the topic. He decides to learn more and wants to create a scatter plot for the data pairs where is the age in years and the distance of the visual near point in centimeters. Then, he needs to write an exponential model for the original data pairs Give Ignacio a hand.

Answer

Scatter Plot:

answer

Example Exponential Model:

Hint

To create the scatter plot make a table of values and plot the obtained points on an 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 will be created using the given information.

Next, the obtained points will be plotted on an coordinate plane. Since all the values are positive, only the first quadrant will be considered.

obtained points

The points are not collinear. However, they lie close to a straight line. Consider the line passing through the first and last points.

line of fit
Since the set of points 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 therefore the form of the equation should be as follows.
Two points can be used to find the slope The points and can be substituted in the Slope Formula.
Evaluate right-hand side

Knowing that the slope is about a partial equation of the line can be written.
To find the value of the intersection with the vertical axis, a point must be used. For simplicity, choose the first point of the table.

Solve for
The equation of the line can now be written.
Finally, to find the exponential model that represents the original data, the variable must be isolated. To do so, the definition of a natural logarithm will be used.
To simplify the obtained equation, the Product of Powers Property and the Power of a Power Property will be used.
Simplify right-hand side
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.
Closure

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.

It was previously discussed that if graphing the data points in the form results in a linear pattern, then an exponential function is a good model for the data points in the form To find the equation of the exponential model using a graphing calculator, press choose EDIT, and enter the values in the first two columns, and

Now, to find the values of place the cursor in the heading of and press After that, press and then before finally closing the parentheses. By pressing will be filled automatically with the corresponding values for

Having entered the values, they can be plotted in a scatter plot by pressing and 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 and as XList and YList, respectively. Any mark can be picked.

After this, press If needed, the scale and the minimum and maximum and values can be adjusted by pressing

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 and from the CALC menu choose the exponential regression option.

Here, and rounded to three decimal places. With this information, the exponential function can be written.
Note that the value of is not the same as in the last example, but the value of is exactly the same.