# Engaging students: Exponential Growth and Decay

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Madison duPont. Her topic, from Precalculus: exponential growth and decay.

How could you as a teacher create an activity or project that involves your topic?

Through an EDSE 4000 assignment (for which we were to find a Higher Level Task,) I found a fantastic activity that demonstrates exponential growth and decay in an exploratory, hands-on manner. The link to the website with the lesson plan as well as the activity can be found below. This activity is beneficial to the students for several reasons. The first is that they use a variety of materials and methods: hands-on manipulatives (M&Ms), technology (graphing calculators), and written work. This provides students with varied learning styles a chance to participate in and understand the concept of exponential growth and decay. Consequently, the students are able to experience how quickly exponential growth and decay occurs as the number of M&Ms they are having to count, collect, shake, and dump on their desk grows or shrinks rapidly. They then are able to see how this real-life phenomenon can be measured mathematically through an equation and represented mathematically in a graph. Another reason why I enjoyed this activity was because the worksheet had them make conjectures, analyze data, and find relationships between factual and actual information. This activity was conducted in my EDSE 4000 class and proved to even interest colleagues because the likelihood of getting an exponential relationship from probability of M&Ms facing a certain way seemed unlikely and intriguing. There were a few tips I took away from conducting the activity in my class that may be helpful to remember when conducting this activity again. First, be sure to instruct students not to eat any of the M&Ms until after they complete both the growth and decay portion. Second, inform students of how to count morphed or faded M&Ms prior to the activity. Third, the students will need to be slightly informed about exponential functions in order to make conjectures or determine theoretical functions as required in the worksheet. Fourth, going over how to use the calculator as directed prior to or during the activity may help the activity run more smoothly. Lastly, skittles do not work as well with this activity because they make a significantly sticky mess as they melt in hands. Overall, the hands-on exploration and intellectual reasoning utilized in this activity makes exponential growth and decay interesting, entertaining, and relatable.

How has this topic appeared in the news?

Exponential growth and decay is largely recognized in the news media regarding the Exponential Growth in Technology. The links below provide intriguing information about the study of how quickly and steadily technology is growing. Morris’ Law is referenced often to provide some explanation for the startlingly rapid growth of technology and decay of previous forms of technology. Also, provided on these sites are videos of Ray Kurzweil discussing his theories of technology being able to duplicate patterns and behaviors of the human brain even more powerfully than that of a human in the near future due to the exponential pattern of technology’s growth. This would likely be interesting to students as technology is a growing part of their lives, lives that may become even more dependent on technology in this coming generation’s lifetime. All of this plausible reality being convincingly calculated from a simple exponential pattern that can be introduced in a high school classroom is pretty amazing, and possibly even powerful, to the minds of future students that can apply this knowledge to the technology phenomenon (or maybe even in other topics of our society) in their future careers. Another video found on the thatsreallypossible.com site has Dr. Albert Bartlett discussing the relevance and impact of “simple” exponential relationships applied to our global community’s resources and economy that are not just hypothetical, but that have happened, and are likely to happen. Using these sites you not only show students the power and importance of exponential growth and decay, you also inform them as global citizens and expose them to realistic problems and ideas that will need to be solved or explored in their lifetime or near future, which is arguably the essence of teaching.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The graphing applet found at mathwarehouse.com (referenced below) is extremely useful in extending student knowledge of the principles of exponential growth and decay. Using this for an activity can help students compare and contrast changing elements of the function without working separate (seemingly unrelated) examples on their own or in groups. Not only is the applet beneficial because you can graph several factors at a time, but you also have clear, graphical representation of the algebraic manipulations along side the algebra. This can be useful for students that learn visually or are ELLs. Activities can be easily carried out by projecting the applet onto a SMART board for full-class evaluation and discussion, having students perform exercises in groups and recording findings for notes, or even just helping students understand differences in homework problems, and hard to understand textbooks notation that are not making sense to students with verbal or written explanations. This being a free website students can access at home on their computer, smart phone, tablet, etc. can be resourceful to students that do not have a graphing calculator and can also be helpful to students as they work through problems independently and try to understand the behaviors of exponential growth and decay outside of the classroom. Because of the applet’s accessibility, aesthetic set up, and ease in manipulation, I recommend this as a useful technology resource both for the teacher and the student as they explore exponential growth and decay.

Pleather, D. (n.d.). Precalculus Lesson Plans and Work Sheets. Retrieved November 17, 2016, from http://www.pleacher.com/mp/mlessons/algebra/mm.html.

Document: M&M_GrowthDecayActivity

http://bigthink.com/think-tank/big-idea-technology-grows-exponentially

http://www.thatsreallypossible.com/exponential-growth/

http://www.mathwarehouse.com/exponential-growth-and-decay/interactive-exponential-graph-applet.php

# Engaging students: Half-life

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Brianna Horwedel. Her topic: working with the half-life of a radioactive element.

How can this topic be used in your students’ future courses in mathematics or science?

Half-Life of radioactive elements in Pre-calculus is generally used when introducing exponential decay. However, its main application is in the field of Chemistry and Archeology. If students go on to take any type of chemistry, they will definitely learn more about the half-life of radioactive elements and how long it takes to get rid of certain nuclear elements. The half-life of Carbon-14 is especially important in Archeology. Carbon-14 dating is a method used to determine the age of archeological artifacts of a biological origin using the half-life of Carbon-14. This process can date bone, wood, cloth, plant fibers, and more that are up to 50,000 years old. The way it works is as follows: as soon as a living organism dies, it stops taking in new carbon. The ratio of carbon-12 to carbon-14 is the same as every living thing. However, when an organism dies, the carbon-14 starts decaying with its half-life of 5,700 years. The carbon-12 does not decay. When an organism is found, they look at the ratio of carbon-12 to carbon-14 to determine the age based on the half-life of carbon-14.

How could you as a teacher create an activity or project that involves your topic?

I think this topic lends itself nicely to a project. Firstly, I would come up with several half-lives and place them in a bowl. Each student would pick a half-life and have to make up an element. Using poster-board, they would give a brief description of what their element is and then create a graph illustrating their particular half-life. They would then present it to the class explaining how they graphed their line and what equation they used. They could also include a table of input and output values. This would be a great refresher on graphing exponential decays along with allowing a little creativity. I think the students would have a lot of fun with this type of project.

How can technology be used to effectively engage students with this topic?

I found this really great web-site (https://jeopardylabs.com/play/exponential-growth-decay) that has an exponential growth and decay form of Jeopardy. It allows you to pick how many teams there are and then it sets up a Jeopardy board. This would be a really fun way to review at the end of a unit over exponential growth and decay. To make the students more engaged, I would offer extra credit to the team with the highest score at the end. Because it is in a game form, students are more likely to pay attention to this type of review.

# Applying Science to Speed Training

I enjoyed this surprising (well, surprising to me) application of exponential functions: training sprinters and other runners.

# My Favorite One-Liners: Part 82

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In differential equations, we teach our students that to solve a homogeneous differential equation with constant coefficients, such as

$y'''+y''+3y'-5y = 0$,

the first step is to construct the characteristic equation

$r^3 + r^2 + 3r - 5 = 0$

by essentially replacing $y'$ with $r$, $y''$ with $r^2$, and so on. Standard techniques from Algebra II/Precalculus, like the rational root test and synthetic division, are then used to find the roots of this polynomial; in this case, the roots are $r=1$ and $r = -1\pm 2i$. Therefore, switching back to the realm of differential equations, the general solution of the differential equation is

$y(t) = c_1 e^{t} + c_2 e^{-t} \cos 2t + c_3 e^{-t} \sin 2t$.

As $t \to \infty$, this general solution blows up (unless, by some miracle, $c_1 = 0$). The last two terms decay to 0, but the first term dominates.

The moral of the story is: if any of the roots have a positive real part, then the solution will blow up to $\infty$ or $-\infty$. On the other hand, if all of the roots have a negative real part, then the solution will decay to 0 as $t \to \infty$.

This sets up the following awful math pun, which I first saw in the book Absolute Zero Gravity:

An Aeroflot plan en route to Warsaw ran into heavy turbulence and was in danger of crashing. In desparation, the pilot got on the intercom and asked, “Would everyone with a Polish passport please move to the left side of the aircraft.” The passengers changed seats, and the turbulence ended. Why? The pilot achieved stability by putting all the Poles in the left half-plane.

# My Favorite One-Liners: Part 42

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

The function $f(x) = a^x$ typically exhibits exponential growth (if $a > 1$) or exponential decay (if $a < 1$). The one exception is if $a = 1$, when the function is merely a constant. Which often leads to my favorite blooper from Star Trek. The crew is trying to find a stowaway, and they get the bright idea of turning off all the sound on the ship and then turning up the sound so that the stowaway’s heartbeat can be heard. After all, Captain Kirk boasts, the Enterprise has the ability to amplify sound by 1 to the fourth power.

# Engaging students: Introducing the number e

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Loc Nguyen. His topic, from Precalculus: introducing the number $e$.

How could you as a teacher create an activity or project that involves your topic?

To be able to understand where the number e is produced in the first place, students need to understand how compound interest is calculated.  Before introducing the number e, I will definitely create an activity for the students to work on so that they can eventually find the formula for compounding interest based on the patterns they produce throughout the process.  The compound interest formula is F=P(1+r/n)nt.  From this formula, I will again provide students a worksheet to work on.  In this worksheet, I will let P=1, r=100%, t=1, then the compound interest formula will be F=(1+1/n)n. Now students will compute the final value from yearly to secondly.

When they do all the computation, they will see all the decimal places of the final value lining up as n gets big.  And finally, they will see that the final value gets to the fixed value as n goes to infinity.  That number is e=2.71828162….,

How has this topic appeared in the news?

To help the students realize how important number e is, I would engage them with the real life examples or applications. There were some news that incorporated exponential curves. First, I will show the students the news about how fast deadly disease Ebola will grow through this link http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary.  The students will eventually see how exponential curve comes into play. After that I will provide them this link, http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/, in this link, the article talked about the global population rate and it provided the scientific evidence that showed the data collected represent the exponential curve.  Up to this point, I will show the students that the population growth model is:

Those examples above was about the growth.  For the next example, I will ask the students that how the scientists figured out the age of the earth.  In this link, http://earthsky.org/earth/how-old-is-the-earth, the students will learn that the scientists used Modern radiometric dating methods to calculate the age of earth.  At this time, I will show them radioactive decay formula and explain to them that this formula is used to determine the lives of the substances such as rocks:

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

To introduce to the students what the number e is, I will engage them with two videos. In the first video, https://www.youtube.com/watch?v=UFgod5tmLYY, the math song “e a magic number” will engage the students why it is a magic number.  While watching this clip, the students will be able to learn the history of e.  Also the students will see many mathematical formulas and expressions that contain e.  This will give them a heads up that they will see these in future when they take higher level math.  It is also pretty humorous of how Dr. Chris Tisdell sang the song.

In the second video, https://www.youtube.com/watch?v=b-MZumdfbt8, it explained why e is everywhere.  The video used probability and exponential function to illustrate the usefulness of e, and showed how e is involving in everything.  It gave many examples of e such as population, finance…  Also the video illustrates the characteristics of the number e and the function that has e in it.  Watching these videos will enhance students’ perception and understanding on the number e, and help them to see how important this number is.

Reference

http://www.math.unt.edu/~baf0018/courses/handouts/exponentialnotes.pdf

http://cleantechnica.com/2014/07/22/exponential-growth-global-solar-pv-production-installation/

http://www.npr.org/sections/goatsandsoda/2014/09/18/349341606/why-the-math-of-the-ebola-epidemic-is-so-scary

http://earthsky.org/earth/how-old-is-the-earth

# Engaging students: Graphing exponential growth and decay functions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Irene Ogeto. Her topic, from Precalculus: graphing exponential growth and decay functions.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The Legend of the Chessboard is a famous legend that illustrates exponential growth. A courtier presented a Persian king with the chessboard and as a reward the courtier asked the king for a grain of rice in each square of the chessboard, doubling the amount in each new square. The king agreed and gave the courtier 1 grain of rice in the first square, 2 grains of rice in the second, four grains of rice in the third and so on. The king didn’t realize how rapidly the amount of grain of rice would grow in each square. This video would be a great way to engage the students into the topic at the beginning of the lesson. The Legend of the Chessboard shows how rapidly exponential functions can grow. After watching the video the students can try to guess or calculate the total number of grains of rice the courtier would get in the end. Afterwards, the students can then graph the exponential function.

The students can use this website to check their guess:

http://britton.disted.camosun.bc.ca/jbchessgrain.htm

How could you as a teacher create an activity or project that involves your topic?

In order to explore graphing exponential growth and decay functions, the students could play a card sort matching game. The students will work in groups to play the card sort matching game. Some students will be given the graphs and have to use the points given to derive the equation. Some groups will be given the equations and have to create the graphs of the exponential functions. As a class, we will go over graphing exponential growth and decay functions and analyze the graphs. The students will be expected to identify the domain, range, asymptotes, y-intercepts and whether the graph is exponential growth or exponential decay. Also, we could explore how exponential functions compare to other functions that we previously studied. This is a great activity that can be used as review before an exam.

How can this topic be used in your students’ future courses in mathematics or science?

Exponential functions are used to model different real world scenarios involving population, money, finances, bacteria and much more. Students can use exponential functions in other courses such as Calculus, Biology, Chemistry, Physics, and Economics. In calculus, students explore differentiation and integration of exponential functions. Given the position of an object in exponential form, students can use Calculus to determine if the object will stop moving.  Newton’s Law of Cooling is an example in physics that demonstrates exponential decay. Compound interest is a major application of exponential functions in finances. Exponential population growth, carbon dating, pH and concentrations of drugs are other examples in math and science that can be modeled by exponential growth and decay functions. In addition, students explore logarithmic functions, the inverses of exponential functions. Being able to recognize and graph exponential growth and decay functions is an important concept that can help students’ in their future courses in math or science.

References:

http://britton.disted.camosun.bc.ca/jbchessgrain.htm

http://www.shsu.edu/kws006/Precalculus/3.2_Applications_of_Exponential_Functions_files/3.2%20Applications%20of%20Exponential%20Functions%20(slides%204%20to%201).pdf

# Engaging students: Exponential Growth and Decay

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Chais Price. His topic, from Precalculus: exponential growth and decay.

How does this topic extend what your students should have learned in previous courses?

Every year in elementary through high school it seemed like I had some form of standardized test. These test typically consist of various problems, which include patterns and sequences of patters style of problems. I always found it more helpful when being introduced to more complex and intimidating concepts, to relate the general idea to something much more simplistic. When teaching a lesson on exponential growth and/or decay I plan on starting off the lesson with problems like:

These two patterns are pretty basic and finding the next one in the sequence shouldn’t be to difficult. This begs the question what if I wanted to find some enormous value for n. For questions d, a student can answer the question by drawing or counting but it will take some time. Or the student could find an equation that models the sequence of patterns. The equation would obviously be an exponential. From this point the teacher could discuss how these functions appear on the graph by simply observing what is happening in the sequence. In the first picture alone with the triangles, we only have 4 triangles shown and the first triangle is solid black. If we continue on, the next one in the sequence would represent basically our x values on a graph and the amount of triangles growing exponentially represents the y values. By using this previous knowledge the teacher was capable of relating a new concept with a much simpler approach.

How could you as a teacher create an activity or project that involves your topic?

Has anyone ever asked you if you would rather have a million dollars, or a penny that doubles everyday for an entire month? I heard this question probably when I was in high school. I am pretty sure that I picked a penny that doubled everyday for a month only because it was the least obvious and it seemed like a trick question. However this is an example of how only 31 days explode into a fortune. After the first week of doubling you only have a little over a dollar. In fact you really don’t start making any real money until about the middle of the 3rd week if you chose to have a penny double everyday for a month. It turns out that by the last day of the month you end up with over 21 million dollars. This is once again because the function is growing exponentially. The link at the bottom of the page has a story that uses this same idea about a raja from India who made a young girls request to have a grain of rice double everyday for a month. This story can be fun to read and engaging for the students as well. After the story is read, there is a calendar where the students will fill in each day the amount of rice given to Rani, the young girl in the story. This calendar has a few random days filled in so the students know if they are on the right track. This activity serves as an engage/ explore for more of an introduction to exponential growth. The students could graph this function of type some points into the calculator to see the function explode. Let x represent days and y represent the grain of rice each day.

How can technology be used to effectively engage students with this topic?

Dan Myers is a teacher who developed a style of teaching called the 3 act lessons, which incorporates multiple technology applications such as video recording, as well as imaging and photo editing. Each act is designed to teach a lesson like a movie divided up into parts. I came across this lesson of his which I think is awesome. In act 1, there is a 24 second video with these words at the beginning: “ a smaller domino can topple a domino that is up to 1.5 times larger in every dimension. “ The guy on the video explains that the smallest domino is 5 mm high and 1mm thick. This is all you are given. Then the teacher asks something to the class along the lines of “ If you wanted to topple over a domino the size of a sky scraper, how many dominoes would you need? “ This opens the door for students to both question and reason. Make a prediction and write it down. Have the students write down an answer they know is too high and one they know is too low. That is the end of act one. As we get into act to we need more information just like in a movie. Act 2 answers the question how many dominoes are present in the video. It also provides a data sheet that has the heights f several sky scrappers. This is a very discussion style lesson so in act 2 we would continue to promote discussion and questions. Then finally in act three we come to the conclusion. The man in the video had 13 dominoes and the biggest one was barely up to his waste. It turns out that if we were to keep adding dominoes that grew 1.5 times more than the previous one, the 29th domino would be as tall as the Empire State Building. That is exponential growth at its finest.

# Calculators and Complex Numbers: Index

I’m using the Twelve Days of Christmas (and perhaps a few extra days besides) to do something that I should have done a long time ago: collect past series of posts into a single, easy-to-reference post. The following posts formed my series on how the trigonometric form of complex numbers, DeMoivre’s Theorem, and extending the definitions of exponentiation and logarithm to complex numbers.

Part 1: Introduction: using a calculator to find surprising answers for $\ln(-5)$ and $\sqrt[3]{-8}$. See the video below.

Part 2: The trigonometric form of complex numbers.

Part 3: Proving the theorem

$\left[ r_1 (\cos \theta_1 + i \sin \theta_1) \right] \cdot \left[ r_2 (\cos \theta_2 + i \sin \theta_2) \right] = r_1 r_2 (\cos [\theta_1+\theta_2] + i \sin [\theta_1+\theta_2])$

Part 4: Proving the theorem

$\displaystyle \frac{ r_1 (\cos \theta_1 + i \sin \theta_1) }{ r_2 (\cos \theta_2 + i \sin \theta_2) } = \displaystyle \frac{r_1}{r_2} (\cos [\theta_1-\theta_2] + i \sin [\theta_1-\theta_2])$

Part 5: Application: numerical example of De Moivre’s Theorem.

Part 6: Proof of De Moivre’s Theorem for nonnegative exponents.

Part 7: Proof of De Moivre’s Theorem for negative exponents.

Part 8: Finding the three cube roots of -27 without De Moivre’s Theorem.

Part 9: Finding the three cube roots of -27 with De Moivre’s Theorem.

Part 10: Pedagogical thoughts on De Moivre’s Theorem.

Part 11: Defining $z^q$ for rational numbers $q$.

Part 12: The Laws of Exponents for complex bases but rational exponents.

Part 13: Defining $e^z$ for complex numbers $z$

Part 14: Informal justification of the formula $e^z e^w = e^{z+w}$.

Part 15: Simplification of $e^{i \theta}$.

Part 16: Remembering DeMoivre’s Theorem using the notation $e^{i \theta}$.

Part 17: Formal proof of the formula $e^z e^w = e^{z+w}$.

Part 18: Practical computation of $e^z$ for complex $z$.

Part 19: Solving equations of the form $e^z = w$, where $z$ and $w$ may be complex.

Part 20: Defining $\log z$ for complex $z$.

Part 21: The Laws of Logarithms for complex numbers.

Part 22: Defining $z^w$ for complex $z$ and $w$.

Part 23: The Laws of Exponents for complex bases and exponents.

Part 24: The Laws of Exponents for complex bases and exponents.

# Inverse Functions: Logarithms and Complex Numbers (Part 30)

Ordinarily, there are no great difficulties with logarithms as we’ve seen with the inverse trigonometric functions. That’s because the graph of $y = a^x$ satisfies the horizontal line test for any $0 < a < 1$ or $a > 1$. For example,

$e^x = 5 \Longrightarrow x = \ln 5$,

and we don’t have to worry about “other” solutions.

However, this goes out the window if we consider logarithms with complex numbers. Recall that the trigonometric form of a complex number $z = a+bi$ is

$z = r(\cos \theta + i \sin \theta) = r e^{i \theta}$

where $r = |z| = \sqrt{a^2 + b^2}$ and $\tan \theta = b/a$, with $\theta$ in the appropriate quadrant. This is analogous to converting from rectangular coordinates to polar coordinates.

Over the past few posts, we developed the following theorem for computing $e^z$ in the case that $z$ is a complex number.

Definition. Let $z = r e^{i \theta}$ be a complex number so that $-\pi < \theta \le \theta$. Then we define

$\log z = \ln r + i \theta$.

Of course, this looks like what the definition ought to be if one formally applies the Laws of Logarithms to $r e^{i \theta}$. However, this complex logarithm doesn’t always work the way you’d think it work. For example,

$\log \left(e^{2 \pi i} \right) = \log (\cos 2\pi + i \sin 2\pi) = \log 1 = \ln 1 = 0 \ne 2\pi i$.

This is analogous to another situation when an inverse function is defined using a restricted domain, like

$\sqrt{ (-3)^2 } = \sqrt{9} = 3 \ne -3$

or

$\sin^{-1} (\sin \pi) = \sin^{-1} 0 = 0 \ne \pi$.

The Laws of Logarithms also may not work when nonpositive numbers are used. For example,

$\log \left[ (-1) \cdot (-1) \right] = \log 1 = 0$,

but

$\log(-1) + \log(-1) = \log \left( e^{\pi i} \right) + \log \left( e^{\pi i} \right) = \pi i + \pi i = 2\pi i$.

This material appeared in my previous series concerning calculators and complex numbers: https://meangreenmath.com/2014/07/09/calculators-and-complex-numbers-part-21/