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.