In Exercises 1–40, use properties of logarithms to expand each lo...

Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

log5 ∛((x^2 y)/24)

Accepted Answer

everyone in this problem were asked to expand And try to evaluate the logarithmic expression were given K. Using our properties of logs and the expression were given is log base two Of the square root of a cubed b squared over 18. True. So we're starting here now let's recall that if we have the end route of em we can write this as M to the one over end. Okay. So that's gonna allow us to write that square root as a cubed B squared over 18 to the exponent. 1/2. Alright now we're calling our log rules. When we have an exponent inside of our log, what can we do? Well if we have log base B of X to the end we can bring the end the exponent outside of the log and multiply it. So we get n log base B of X. So in our case the one half the exponent comes out and multiply. So we get one half log base two of a cube B squared over 18. All right now we have in our log one term divided by another. What's the log rule for dealing with divisions like this? Within our log? If we have log base B some X over some Y. Okay, we can turn this into a subtraction of two logs. So we have log base B of X minus log base B. Of why? Okay. So in our case we have one half. Okay and be careful here this one half multiplies to this entire logarithms. Okay. So it's gonna have to multiply to both of the terms and the subtraction. And that's that's gonna be a common mistake. Mr equal center. That can be a common mistake where you don't have that one half applied to both terms. Okay so put a big square bracket here that will just help remind you that one half applies to both terms is log base two of a Q. B squared in the numerator. And then you're going to subtract log base two of the denominator. Alright so we have base B. We've kept the same base in both of our logs. We have the numerator subtracted by the log of the denominator. Great. What can we do now? Well now we have this term. Okay we have this A Q. B squared and those are being multiplied. What does our log will tell us if we have something multiplied within our log we're gonna write the log rule on this side. If we have log base B. Of X times Y. We can write log base B. Of X. Plus log base B. Of what case? It's very similar to our subtraction role. Instead of subtracting we're going to add. So we end up with 1/2 big square bracket log base two of a cute plus log base two of B squared. Okay. They were being multiplied within the log. Now there are two separate logs being added together And we still have our subtraction minus log base two of 18. Alright great. Let's go ahead and Band or 1/2. Okay so we're gonna have 1/2 log base two have a cubed plus one half log base two B squared minus one half log base two of 18. Okay. And you can choose to do that expanding a little bit later if you wanted. But we're gonna go ahead and do it there. Okay and then let's continue. Alright so now we have exponents here. Okay so we have the exponent with a cube exponents B square. Now let's remember what we did at the beginning of the problem when we have an exponent. Okay here we have an exponent of one half and we used our exponent rule here. Okay so we're going to apply that same rule again. You'll see that in these problems, you often have to apply the same rule more than once. We're going to apply that same rule again to both of these terms. So the three the exponent of three here on the A. That's going to come out in front. That's gonna give us 3/2 log base two of a. Okay, similarly the two is going to come, the exponent of two. That's with the B. Is going to come out in front two divided by two is just going to be one. So we're gonna be left just with log base two of B. Okay now that's true. Let's see what we can do here. We know if we have log base two of two, That's going to equal one case. A log where the base and the thing inside the log are the same is going to give us one. So let's see if we can rewrite this with it too. We know 18 is going to be two times nine. Okay. Now we have a multiplication. What do we do when we have a multiplication? Well this is the rule we use here. The multiplication rule we used earlier to separate the cube and the B squared. We're gonna use it again to separate the two and the nine. Okay. So these terms at the beginning we're gonna leave alone three over to log base two of a. Okay, this is as simple or as expanded as we can get this term. There's nothing more we can do here. Same with the log base two of B term. Nothing more that we can expand here. Okay now we have minus one half and again big square bracket. That minus one half is going to apply to both terms big square brackets. So we don't forget that. Yes We're gonna get log base two up to Plus Log Base two of 9. Okay, that's according to that multiplication rule over here. Alright now we have our log base two of two. We know if we have log base with whatever's inside the log the same, that's gonna be one. Okay so we can simplify this three half, Log. Base two of a Plus Log Base two B. Okay We're gonna have -1/2. This is just going to be one And then our log base two of 9. All right so one last step scroll down a little bit more is just to expand that minus one half at the end. So we're gonna get three half log base two of a Last Log Base two of B minus one half minus one half log base two of nine. Now looking here there's one more thing we can do, we see that we have a nine here. Okay we can write that simpler right? We can reduce nine down to something simpler. We have three squared. Alright let's keep going. So three halves Log Base two and again we're being asked to expand and and evaluate as much as we can. So when we see something like that we want to go ahead and do that Log Base two B minus one half. And then we have minus one half log base two of three squared. Alright now we have an exponent in the log rhythm again that looks just like these other blue terms we've dealt with. We're gonna use that rule we can bring the exponent down and in front. And what that's gonna allow us to do is have the coefficient of one. Okay so the two is going to come down in front two divided by two is gonna be one. We're just gonna be left with log base two of three. Alright, so that looks good. We can't expand anything or evaluate anything else. So let's go ahead. This is our solution and just compare this with the answer choices that were given. So between those answer choices we are going to have C three log base two of a divided by two plus three. Sorry? Plus log base two of b minus log base two of three minus one half. Okay. Alright. Thanks everyone for watching see you in the next video.

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log5 ∛((x^2 y)/24)

Key Concepts

Properties of Logarithms

Radicals and Exponents

Evaluating Logarithmic Expressions

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