# Lesson 3 Chopping Logs Solidify Understanding

## Jump Start

### 1.

Graph the equation:

### 2.

Write the equation for the base

## Learning Focus

Use graphs to discover properties of logarithms.

Justify conjectures about logarithms.

Can the graphs of logarithmic functions give us insight into some of the algebraic properties of logarithms?

## Open Up the Math: Launch, Explore, Discuss

Abe and Mary are working on their math homework together when Abe has a brilliant idea.

**Abe**: I was just looking at this log function that we graphed in *Falling Off A Log*:

I started to think that maybe I could just “distribute” the log so that I get:

I guess I’m saying that I think these are equivalent expressions, so I could write it this way:

**Mary**: I don’t know about that. Logs are tricky and I don’t think that you’re really doing the same thing here as when you distribute a number.

### 1.

What do you think? How can you verify if Abe’s idea works?

### 2.

If Abe’s idea works, give some examples that illustrate why it works. If Abe’s idea doesn’t work, give a counterexample.

**Abe**: I just know there is something going on with these logs. I just graphed

It’s weird because I think this graph is just a translation of

### 3.

How would you answer Abe’s question? Are there conditions that could allow the same graph to have different equations?

**Mary**: When you say, “a translation of

### 4.

Find an equation for

**Mary**: I wonder why the vertical shift turned out to be up

### 5.

Try to write an equivalent equation for each of these graphs that is a vertical shift of

#### a.

Equivalent equation:

#### b.

Equivalent equation:

**Mary**: Oh my gosh! I think I know what is happening here! Here’s what we see from the graphs:

Here’s the brilliant part: We know that

I think it looks like the “distributive” thing that you were trying to do, but since you can’t really distribute a function, it’s really just a log multiplication rule. I guess my rule would be:

### 6.

How can you express Mary’s rule in words?

### 7.

Is this statement true? If it is, give some examples that illustrate why it works. If it is not true provide a counterexample.

**Mary**: So, I wonder if a similar thing happens if you have division inside the argument of a log function. I’m going to try some examples. If my theory works, then all of these graphs will just be vertical shifts of

### 8.

Here are Mary’s examples and their graphs. Test Mary’s theory by trying to write an equivalent equation for each of these graphs that is a vertical shift of

#### a.

Equivalent equation:

#### b.

Equivalent equation:

### 9.

Use these examples to write a rule for division inside the argument of a logarithm that is like the rule that Mary wrote for multiplication inside a logarithm.

### 10.

Is this statement true? If it is, give some examples that illustrate why it works. If it is not true provide a counterexample.

**Abe**: You’re definitely brilliant for thinking of that multiplication rule. But I’m a genius because I’ve used your multiplication rule to come up with a power rule. Let’s say you start with:

Really that’s the same as having:

So, I could use your multiplying rule and write:

I notice there are 3 terms that are all the same. That makes it:

So my rule is:

If your rule is true, then I have proven my power rule.

**Mary**: I don’t think it’s really a power rule unless it works for any power. You only showed how it might work for 3.

**Abe**: Oh, good grief! Ok, I’m going to say it can be any number,

Are you satisfied?

### 11.

Provide an argument about Abe’s power rule. Is it true or not?

**Abe**: Before we win the Nobel Prize for mathematics, I suppose that we need to think about whether or not these rules work for any base.

### 12.

The three rules, written for any base

**Logarithm of a Product Rule: **

**Logarithm of a Quotient Rule: **

**Logarithm of a Power Rule: **

Make an argument for why these rules will work in any base

## Ready for More?

Find an example that demonstrates the logarithm rules in bases other than base

## Takeaways

Properties of Logarithms:

## Lesson Summary

In this lesson, we examined graphs to find equivalent expressions for logarithmic functions. We justified three logarithm properties that are true for any logarithm base. They are:

**Logarithm of a Product Rule: **

**Logarithm of a Quotient Rule: **

**Logarithm of a Power Rule: **

### 1.

Rewrite the expression using exponents.

### 2.

Rewrite

### 3.

Rewrite the equation