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Combine the given expression into a single logarithm.$$4 \log x-2 \log y+1 / 2 \log z-\frac{2}{3} \log w$$

$$\log \frac{x^{4} \sqrt{z}}{y^{2} \sqrt[3]{w^{2}}}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

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02:10

Combine the given expressi…

00:58

Write each logarithmic exp…

01:15

Simplify the expression.

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01:28

Condense the expression to…

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Rewrite each expression as…

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00:56

03:43

Write the expression as on…

02:42

01:53

All right. So we're we have this expanded form. We have four log of X and minus to log of life because plus one half law busy. Okay, Minus two thirds log of w eso. Basically, we're gonna do to Law of logs, I guess, technically, three. But that's okay. Um is the first one is I'm just moving these exponents. Now, if you only want to do to love logs, then what you can do. And so I'm going to show you what I typically teach. You don't have to do it my way. Is that x to the fourth and just make the entire thing log of y to the negative second power, Um, plus log of Z to the one half and then plus log of W to the negative two thirds. So what I'm doing, I'm attaching the negative to that term attached to the negative because then I only have to use one more law of logs. And that's combining things using multiplication because they're all edition now X to the fourth. Why do the next second z to the one half and w to the neck of two thirds? And now all you have to do is think about well, I think two things I think I could do this all at once is ah that I have X to the fourth and negative Expo is gonna move to the denominator the Z to the one half. That means the square root of the notice. It's still in the numerator, though, because it's positive. And then, ah, negative exponents for W goes into the denominator. That means the cube root because the third is in the denominator and then squared. It doesn't matter if you put the squared inside on the outside. You don't have to do the problem this way. What you could have done instead is leave Jesus negatives and anything that has a negative gets combined with division. Now my students struggle with that because they really struggled. They think the entire thing from that point forward is in the denominator. Um, that's what I like to teach it this way that Onley the negative pieces get put into the denominator eso. Here's your final answer

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