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### Course: Algebra 2 > Unit 8

Lesson 2: Properties of logarithms- Intro to logarithm properties (1 of 2)
- Intro to logarithm properties (2 of 2)
- Intro to logarithm properties
- Using the logarithmic product rule
- Using the logarithmic power rule
- Use the properties of logarithms
- Using the properties of logarithms: multiple steps
- Proof of the logarithm product rule
- Proof of the logarithm quotient and power rules
- Justifying the logarithm properties

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# Justifying the logarithm properties

Study the proofs of the logarithm properties: the product rule, the quotient rule, and the power rule.

In this lesson, we will prove three logarithm properties: the product rule, the quotient rule, and the power rule. Before we begin, let's recall a useful fact that will help us along the way.

In other words, a logarithm in base $b$ reverses the effect of a base $b$ power!

Keep this in mind as you read through the proofs that follow.

## Product Rule: ${\mathrm{log}}_{b}(MN)={\mathrm{log}}_{b}(M)+{\mathrm{log}}_{b}(N)$

Let's start by proving a specific case of the rule — the case when $M=4$ , $N=8$ , and $b=2$ .

Substituting these values into ${\mathrm{log}}_{b}(MN)$ , we see:

And so we have that ${\mathrm{log}}_{2}(4\cdot 8)={\mathrm{log}}_{2}(4)+{\mathrm{log}}_{2}(8)$ .

While this only verifies one case, we can follow this logic to prove the product rule in general.

Notice, that writing $4$ and $8$ as powers of $2$ was key to the proof. So in general, we'd like $M$ and $N$ to be powers of the base $b$ . To do this, we can let $M={b}^{x}$ and $N={b}^{y}$ for some real numbers $x$ and $y$ .

Then by definition, it is also true that ${\mathrm{log}}_{b}(M)=x$ and ${\mathrm{log}}_{b}(N)=y$ .

Now we have:

## Quotient Rule: ${\mathrm{log}}_{b}\left({\displaystyle \frac{M}{N}}\right)={\mathrm{log}}_{b}(M)-{\mathrm{log}}_{b}(N)$

The proof of this property follows a method similar to the one used above.

Again, if we let $M={b}^{x}$ and $N={b}^{y}$ , then it follows that ${\mathrm{log}}_{b}(M)=x$ and ${\mathrm{log}}_{b}(N)=y$ .

We can now prove the quotient rule as follows:

## Power Rule: ${\mathrm{log}}_{b}({M}^{p})=p{\mathrm{log}}_{b}(M)$

This time, only $M$ is involved in the property and so it is sufficient to let $M={b}^{x}$ , which gives us that ${\mathrm{log}}_{b}(M)=x$ .

The proof of the power rule is shown below.

Alternatively, we can justify this property by using the product rule.

For example, we know that ${\mathrm{log}}_{b}({M}^{p})={\mathrm{log}}_{b}(M\cdot M\cdot \text{\u2026}\cdot M)$ , where $M$ is multiplied by itself $p$ times.

We can now use the product rule along with the definition of multiplication as repeated addition to complete the proof. This is shown below.

And so you have it! We have just proven the three logarithm properties!

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