for-the-following-exercises-expand-each-logarithm-as-much-as-possible-rewrite-each-expression-as-a-3

Question

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
$$
\log _{b}\left(\frac{13}{17}\right)
$$

Lucas Everson · Accepted Answer

Hello. Welcome to the video and today we&#x27;re to be taking a look at logger rhythms. This problem were given log base B. Which beacon really equal anything in this situation? It doesn&#x27;t really matter of 13 over 70. The problem tells us to rewrite this Now How exactly we do this, we need to know our log properties. There are quite a few long properties that we know of, but essentially what they do is that they allow us to take logs and rewrite them into different looking logs. But they both equal the same thing. Then they&#x27;re all really important. This we have to use in this problem is tthe equ ocean where if we have a long face B off any value a over any value, See, we can rewrite this into law Base B. It&#x27;s important that we keep the same face value, whatever&#x27;s on top, minus the long face, be of whatever&#x27;s on bottom and these are interchangeable. We can change a log that looks like this into this, and vice versa. And that&#x27;s what we need to do with this problem. So going back to what we&#x27;re given, log base B 13 over 17 could be transformed into our answer right here. Log base B 13 minus log base be 70

K B · Answer

So this question we have a natural log of three a. B in times five seem whereas toe basically put this in the most expanded form that we can using in this case will use Osama box The reason we use a sum of boxes because we&#x27;re multiplying all the terms inside here. So this is gonna be the same as Ellen. Three. Ate pasta Alla and a plus Allenby plus Ellen five plus Ellen seeing. Basically, the reason we got this is because all these things were being multiplied inside the natural lot. We&#x27;ll meet when we write these separate terms being added together. We use addition because of the properties of water.

Lucas Everson · Answer

Hello. Welcome to the video and there would be taking a look at logger rhythms and specifically how we can use their properties to rewrite them into different logarithms. Now, in this problem, we are given the log. And if it doesn&#x27;t have a base value here, you can just pretend that there&#x27;s a 10 here. So we&#x27;re given the log base 10 of extra the 15th power times. Why did the 13th power divided by sea to the 19th? We&#x27;re told to expand this as much as we can. Now, to do this, we&#x27;ll need to use three different properties of Logar rhythms. You begin, we&#x27;re gonna talk about the quotient rule and that reads that if we have a long that looks like this, we can change it into a lot that looks like this. They&#x27;re going through it. If we have log base, any value be doesn&#x27;t really matter what B equals here of any value a divided by any value. See, we can rewrite that as log base be. Keep in mind that this base value does not change lock based, be off whatever&#x27;s on top My ous log base B of whatever&#x27;s on bottom. We&#x27;re gonna be using that. We&#x27;re also using the power rule that tells us that if we have a log base B of some value a raised to the power of C, we can rewrite that as the exponents is in this case. See, Time&#x27;s the log base be once again, the base value does not change here off whatever&#x27;s on Bob. And in this case, that is a. In addition, we will also use the product. And that tells us that if we have I love that looks like this. So log base, any value be of two values multiplied by each other in this case, A and C. We can rewrite that as the log base B off of one of the values plus the same law based, be off the other value. And this may look kind of familiar because it looks pretty similar to the ocean. So applying these three principles were going back to what we were given in the first place, which is the log extra the 15th times y to the 13th oversee to the 19th and we&#x27;re gonna use the power rule to rewrite these three first. So, with extra the 15th we can throw the the 15 out front and multiply that by the log. You can pretend there&#x27;s a 10 here 15 times the log of X do the same with Why? So throw the 13 out front to get 13 times the log of why and sing with C. We&#x27;re gonna do 19 times the log of Z. Now these two values are being multiplied by each other and as well as we know from the product rule, we&#x27;re gonna add this this together and because that whatever that value equals is being divided by Z to the 19th year. We I can subtract this from this here. And if we do that, your final answer should look like this 15 times the log of X plus 13 times the log of why and that&#x27;s that was our numerator in the original. The original expression we were given minus Z so 19 times the log of Z

Lucas Everson · Answer

Hello. Welcome to the video, and today we will be taking a look at longer rhythms. This problem were given log base B seven x times, too. Why? And this problem tells us to expand this as much as we can. Now, in order to do this, we need to understand our log and for this particular problem will need to know E product. And the product rule allows us to rewrite along that looks like this into this. So if we have a long based be any value be of two things being multiplied together, we can rewrite it as log base B of this case, X plus law based B off the other thing being multiply in this case why? And the only thing to know is that the face value always stays the same. It&#x27;s very important. So I&#x27;m going to the problem. Were given log base B seven x times, too. Why now? Before I started using the long property, I started to look at this a little differently. Started to take it apart a little bit. We can look at this as long base B of seven times x times two times why and that might make it a little easier to see how we would use the product from so transforming this into our final answer. You have logged based B seven plus log base B of X plus log base, be of two plus law based Be of why so we essentially just take each component that&#x27;s being multiplied here, attention to a log base B and add them all up to each other, and that is how the product rule works.

K B · Answer

So in this punch, did we have not for long of eight of the negative second over beating the negative fourth seat of the fifth. So basically, we&#x27;re gonna be rewriting. This is not trouble aggravated than the second. And now, instead of writing like the over one of the other terms were instead, what we&#x27;re gonna do is we&#x27;re going to have chopped them because basically, when you do thigh inside this bomber them that&#x27;s the same Mr Trump day, another longer, them without saying about you. So we&#x27;re gonna be subtracting island, beat a negative forth, and we&#x27;re gonna be subtracting natural long of seat of the fifth. But before we are, say that says our final answer, we&#x27;re gonna take these exponents and we&#x27;re gonna make the multipliers of the law algorithm because basically, if you have this term here, if it&#x27;s to an expert in this experiment, can become like a coefficient or multiply so we can make this negative to l a minus, and this would be minus negative for so we could just call this plus four B and then this would meet minus Alan. So this would be minus five out. Let&#x27;s see. And this, we can say, is our final

K B · Answer

So in this question, were asked to rewrite Long based for of Axel Jersey over W in a subtraction of logs in this case. So we&#x27;re gonna have long based four backs minus long base for Z. The reason you would use such options because we&#x27;re dividing exp I see minus long based for W and this is because we multiply. We divide all of this by W So this is why we get along best for X minus log base for Izzy Mind saw baseboard that week.

Problem

For the following exercises, expand each logarith…

Problem 5 Easy Difficulty

For the following exercises, expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs.
$$
\log _{b}\left(\frac{13}{17}\right)
$$

Answer

$o g_{b}(13)-\log _{b}(17)$

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$o g_{b}(13)-\log _{b}(17)$

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