Properties of Logarithms

Properties of Logarithms

Let's keep practicing!

Let's refresh some properties!

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Let's get to some practice...

But FIRST!

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As you work through this Quizizz, you'll be practicing all sorts of different logarithm expressions.

• You can do it as many times as you like

• You benefit the most by trying the questions honestly (don't just randomly tap/click or use outside resources!)

Open Ended

Warm Up: Rewrite the equation in exponential form.

$$\log_8\left(64\right)=2$$  

Multiple Choice

Warm Up: Rewrite the equation in logarithmic form.

$$7^b=a$$  

• $$\log_b\left(7\right)=a$$  

• $$\log_7\left(a\right)=b$$  

• $$\log_7\left(b\right)=a$$  

• $$\log_b\left(a\right)=7$$  

Multiple Choice

Warm Up: Rewrite the expression in exponential form.

$$\log_y\left(z\right)=x$$  

• $$y^x=z$$  

• $$x^z=y$$  

• $$y^z=x$$  

• $$z^y=x$$  

Multiple Choice

Why can't your nose be 12 inches long?

• Because it would be a foot.

• Pick the other option you silly goose.

Now that you're warmed up...

• Review those properties! Let's see if we can expand and condense some logarithms.

We can use properties of logarithms to manipulate expressions.

Multiple Choice

$$\log\left(5\right)+\log\left(2\right)\ =\ ...$$  

Or...I could CONDENSE the expression as:

• $$\log\left(7\right)$$

• $$\log\left(3\right)$$

• $$\log\left(2.5\right)$$

• $$\log\left(10\right)$$

It's beneficial to condense this expression!

Of course, this won't always work in our favor...

But being able to manipulate expressions will help simplify the process of some difficult concepts. Think of it like training your brain.

Multiple Select

You ready to practice? Check ALL the boxes to indicate that you are ready:

• I have my log properties nearby.

• I took a deep breath.

• I have desmos.com/scientific open so I can calculate some stuff.

• I told someone near me what my favorite animal is.

• I CAN DO THIS

Multiple Choice

Hooray! LET'S GOOOOOOOO Use the change of base formula to rewrite the logarithm:

$$\log_7\left(8\right)$$  

• $$\frac{\log\left(8\right)}{\log\left(7\right)}$$  

• $$\log\left(\frac{8}{7}\right)$$  

• $$\log\left(\frac{7}{8}\right)$$  

• $$\frac{\log\left(7\right)}{\log\left(8\right)}$$  

Multiple Choice

Expand the logarithm: $$\log\left(5\cdot3\cdot2\right)$$  

• $$\log\left(30\right)$$  

• $$\log\left(5\right)\cdot\log\left(3\right)\cdot\log\left(2\right)$$  

• $$\log\left(15\right)+\log\left(2\right)$$  

• $$\log\left(5\right)+\log\left(3\right)+\log\left(2\right)$$  

Multiple Choice

Condense the logarithm to a single logarithm. $$2\log\left(5\right)-\log\left(6\right)$$  

• $$\log\left(\frac{5^2}{6}\right)$$  

• $$\log\left(11^2\right)$$  

• $$\log\left(30^2\right)$$  

• $$\log\left(\frac{5}{6}\right)^2$$  

How'd that last one go?

When we expand and condense logarithms, we want to do so fully!

Think about your order of operations to help you with your expansions and condensing. Certain operators will happen before others.

Multiple Choice

Condense the logarithm to a single logarithm. $$2\left[\log\left(5\right)-\log\left(6\right)\right]$$  

• $$\log\left(\frac{5^2}{6}\right)$$  

• $$\log\left(11^2\right)$$  

• $$\log\left(30^2\right)$$  

• $$\log\left(\left(\frac{5}{6}\right)^2\right)^{ }$$  

Multiple Choice

Condense to a single logarithm. $$3\log\left(11\right)+\log\left(2\right)$$  

• $$3\log\left(22\right)$$  

• $$\log\left(11^3\right)+\log\left(2\right)$$  

• $$\log\left(11^3\cdot2\right)$$  

• $$\log\left(22^3\right)$$  

Multiple Choice

Expand the logarithm. $$\log\left(\left(xy^5\right)^2\right)$$  

• $$2\log\left(x\right)+5\log\left(y\right)$$  

• $$2\log\left(x\right)+10\log\left(y\right)$$  

• $$10\log\left(x\right)+2\log\left(y\right)$$  

• $$2\log\left(x\right)-10\log\left(y\right)$$  

How'd that last one go?

Shake out those hands, we have a few more to go.

This would be a good time to stand up and stretch!

Multiple Choice

Expand the logarithm: $$\log\left(\frac{x^6}{y^2}\right)$$  

• $$2\log\left(x\right)+12\log\left(y\right)$$  

• $$6\log\left(x\right)+2\log\left(y\right)$$  

• $$12\log\left(x\right)-2\log\left(y\right)$$  

• $$6\log\left(x\right)-2\log\left(y\right)$$  

Multiple Choice

Condense the logarithm. $$\frac{1}{2}\cdot\log\left(a\right)+\frac{1}{2}\log\left(b\right)+\frac{1}{2}\log\left(c\right)$$

• $$\log\left(\left[abc\right]^2\right)$$  

• $$\log\left(\frac{2a}{\frac{2b}{2c}}\right)$$  

• $$\log\left(\sqrt{abc}\right)$$  

• $$\log\left(a\sqrt{bc}\right)$$  

Multiple Choice

Expand the logarithm. $$\log\left(\left[\frac{11}{6^4}\right]^5\right)$$  

• $$5\log\left(11\right)-20\log\left(6\right)$$  

• $$20\log\left(11\right)-20\log\left(6\right)$$  

• $$20\log\left(11\right)-5\log\left(6\right)$$  

• $$5\log\left(11\right)-4\log\left(6\right)$$  

Breathe!

You're done! How'd you do? Reflect on your process today.

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