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The word (almost) every math student hates, “proof”. When teaching a precalculus class I do not introduce the notion of a rigorous mathematical proof, however, I do try to get the students to understand why a particular result is true. I use the phrase “why might this be true” a lot. My goal is to get the class thinking about how we arrive at such a conclusion instead of merely accepting it. This doesn’t always work, in fact it rarely does. Students are too content with just memorizing a formula and knowing how to use it. If it work, it works, who cares why it works. To a certain extent that’s fine, I mean, it is important to know how to apply a result. But my point is this, it’s just as important to ask why something is true and at least try to understand why it is true. Even if you don’t get it completely.

To emphasize my point, I’m trying something new for my class next quarter. In the past when I’ve taught the properties of logarithms, I would simply write a property on the board followed by a brief justification. I had fallen into the mindset of my students. I didn’t care if they knew how to derive the properties of logarithms, I just cared that they could memorize them and know how to use them when solving equations. I know, I’m a hypocrite. For next quarter though, I’ve come up with a worksheet that will combat this.

Section 5.4 Properties of Logarithms

In this worksheet we will derive some important properties of logarithms. These properties will be essential for us when we solve logarithmic and exponential equaitons in section 5.5.

Before we can talk about the properties of logarithms we need to recall a few important facts.

• The definition of a logarithm: $\log_b(x)=y$ is equivalent to $b^y=x$.
• One to one property of logarithms: If $\log_b(x)=\log_b(y)$ then $x=y$.
• One to one property of exponents: If $b^x=b^y$ then $x=y$.

The Inverse Rules

$\log_b(b^x)=\bigstar$

What might $\bigstar$ be? First use the definition of logs to rewrite as an equivalent exponential equation. Then use the one to one property of exponents to find $\bigstar$.

$b^{\log_b(x)}=\bigstar$

What might $\bigstar$ be? First use the definition of logs to rewrite as an equivalent logarithmic equation. Then use the one to one property of logs to find $\bigstar$.

The Logarithm of a Product

$\log_b(MN)=$ ?

To derive a formula we make a substitution. Let $x=\log_b(M)$ and $y=\log_b(N).$

a) Use the definition of logs to rewrite $x=\log_b(M)$ as an equivalent exponential equation. Do the same for $y=\log_b(N)$.

b) Multiply together your results from part (a) and use properties of exponents to simplify.

c) Your result from part (b) should be a single exponential equation. Use the defintion of logs to rewrite that equation as an equivalent logarithmic equation.

d) Your result from part (c) should be a logarithmic equation that has an $x$ and a $y$ in it. Plug in $x=\log_b(M)$ and $y=\log_b(N)$ into that equation.

e) What can you say about $\log_b(MN)=$ ?

The Logarithm of a Power

$\log_b(M^p)=$ ?

To derive a formula we again make a substitution. Let $x=\log_b(M)$.

a) Use the definition of logs to rewrite $x=\log_b(M)$ as a equivalent exponential equation.

b) Your result from part (a) should be a single exponential equation. Raise both side of that equaiton to the $p^{th}$ power.

c) Use the definition of logs to rewrite your result from part (b) into an equivalent logarithmic equation.

d) Your result from part (c) should be a logarithmic equation that has an $x$ in it. Plug in $x=\log_b(M)$ into that equation.

e) What can you say about $\log_b(M^p)=$ ?

The Logarithm of a Quotient

$\log_b\left(\frac{M}{N}\right)=$ ?

a) Use the properties of exponents to rewrite $\log_b\left(\frac{M}{N}\right)$ as the logarithm of a product.

b) Use the product rule to split up your result from part (a).

c) Use the power rule to simplify your result from part (b).

d) What can you say about $\log_b\left(\frac{M}{N}\right)=$ ?

I still don’t care if the students know how to derive these properties. Rather, my goal here is to help them understand why these properties hold. In doing that I hope that they will also remember the properties better.

Check it out here.

Update. November 18. This is a revision of the original post. There is still no conclusion and I don’t know where to put the last paragraph.

This is a very rough draft of my personal statement. I am applying to PhD programs in math education. The one below is for a math ed program at Berkeley. There is no conclusion, I need to add that. Also, it is a bit too long right now. I need to trim some stuff out.

Any and all comments and criticisms are welcomed and encouraged. Thanks in advance to anyone who has any advice, I really appreciate it!

There are two main reasons why I want to pursue a PhD in math education. First, I want to be involved in decision-making concerning equality, reform, and curriculum development for high school math education. Second, I want to better understand issues in undergraduate mathematics education. My experiences as a student and teacher have allowed me to realize these goals and have prepared me for studying math education at the PhD level.

My first experience in math education came as an undergraduate; I worked at UC Irvine’s Center for Educational Partnerships (CFEP) where I assisted in math instruction at Spurgeon Middle School in Santa Ana, California. The goal of CFEP is to increase postsecondary opportunities for California’s educationally disadvantaged students. I participated in CFEP’s Saturday Academy in Mathematics (SAM), an academic enrichment program. The SAM curriculum, written by CFEP and the Irvine Mathematics Project, is interactive instead of lecture based and encourages students to think and work together. Rather than simply teaching math, I want to be a part of designing and implementing new math curricula since I have seen the positive impact it can have on students.

My work as an undergraduate inspired me to pursue a master’s degree in math. As a part of the masters program at Cal Poly, I have the opportunity to teach my own math class. This past year, I have been teaching pre-calculus at Cal Poly. Teaching my own class has allowed me to reach a broader audience of students at a more advanced level. On the other hand, it has also revealed to me certain weaknesses in math education. Many of the students I teach lack the motivation and interest to learn the material. This apathy in many undergraduate math courses concerns me and I want to understand both the cause, and how to better combat it. Moreover, I am interested in the transition from learning high school math to learning college math.

The masters program at Cal Poly has been a good stepping-stone in preparing me for the challenges of a PhD. I know what it takes to succeed in a graduate program. Over the past year, I have learned how to balance teaching, studying, research, and classes. In my first year, I successfully studied for, and passed, a qualifying exam in real analysis. In addition, I have been conducting research with Dr. Vincent Bonini studying the classification of minimal surfaces. As a result of my research, I have authored a brief survey on minimal surfaces, appearing in this year’s edition of the Polymath Newsletter, and I had the privilege of lecturing at the Cal Poly Differential Geometry Seminar. On top of my teaching, exams, and research I have been able to thrive in my classes by maintaining a 3.74 GPA. The masters program has allowed me to hone my studying and researching skills necessary to excel in the PhD program.

Being a senior graduate student this year, I had the opportunity to mentor new TA’s in the Cal Poly Math Department TA mentoring program. I helped new TA’s in developing their syllabi and shared my teaching experiences.