Problem

For the following exercises, condense each expres…

00:54

Problem 23 Medium Difficulty

For the following exercises, condense each expression to a single logarithm using the properties of logarithms.
$$
\log (x)-\frac{1}{2} \log (y)+3 \log (z)
$$

Answer

$\log \left(\frac{x}{\sqrt{y} z^{3}}\right)$

Related Courses

Algebra

College Algebra

Chapter 6

Exponential and Logarithmic Functions

Section 5

Logarithmic Properties

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Video Transcript

Hello. Welcome to the video, and today we'll be taking a look at logger rhythms and particularly how we can use their properties to rewrite them. And this problem we're given the log of X minus 1/2 a log of why, plus three times the log of Z. We're told to condense this into a single log rhythm and how we do this, we'll need to know and use three log arrhythmic properties, ocean rule, the product rule and the power rule, beginning with the quotient rule. If we have long rhythmic expression that looks like this, you can actually rewrite it as this. So starting from the beginning, if we have the log base, any value be it doesn't really matter what B equals of value a minus the log base be when. And one thing that's important to note is that this base value needs to be the same as the 1st 1 of another value. See, that could be rewritten as the log base B. What's again? Same value off whatever came first, divided by whatever came second. We're not of the product rule, it's pretty similar, but if we have the log base once again any value be of value A plus, the long base be same the same base value of C that could be rewritten as the log base B of those two values multiplied by each other. And then finally, the power rule. If we have a log face B once again, any value off any value a being multiplied by, see, let's just call it. See, We can rewrite that as the long base B of a whatever is here, raised to the power off. See whatever's in front here, so moving on to the problem. This is what we're given originally, but first I wanted to take this section of it and condense it into one lager. Them, too, make our next step a lot easier. So if we have 1/2 times the log of why we can notice that we have a constant right here, it's multiplying. It's being multiplied by you. Log of why so we can use the power rule to throw this 1/2 as an export off the Why condensed that into this loss, and it's the same thing here, except with a three. So we throw the three into the power of Z and rewrite this like this. So now that we've done that, we can see that we have two logs being added to each other. That means we can rewrite whatever is inside here as being front and products of each other. And I just changed the 1/2 to a square root those same thing so we can change this blue circle area here simply to the log of the square root of why Times z huge. Now, finally, uh, this is after what we did in the last slide. This is the results of that. We still have a log X in front of that and his log X minus the lock of the square root of white times z cute. And as we can see here, we have two logs that are being subtracted. That means we can use the quo ocean. And that means that whatever's first here goes in the numerator. Whatever's second, it was in the denominator. And this leads just to our final answer. The long the X divided by the square root of Wyatt times C Cute that your final answer

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