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.

logb ((√x y^3)/z^3)

Accepted Answer

Hey everyone in this problem we are asked to expand the given logarithmic expression and try to evaluate by using the properties of logarithms. And the expression were given is log base de. Of the square root of two A. Times B. To the four divided by C. To the exponent nine. So let's go ahead and rewrite what we were given. So we have log base de. Of the square root of two A. Times B. to the four Divided by c. to the exponent nine. Alright so the first thing we wanna do here when we're expanding this, let's deal with this division. You could deal with this in a couple of different ways. Um We have some multiplication going on. Let's deal with the division first. Okay so recall that if we have log base B. Of two things being divided say x divided by Y. So we can write this as log base B. Of x minus log base B. Of y. Okay so a log rhythm of a division can become the subtraction of two separate logs. Okay we're going to keep the same base and then the arguments for the division become the arguments separately in those. In that subtraction. Okay. Alright so applying this to our problem we're gonna have log base D. Okay we keep the same base of the numerator And our numerator in this case is the square root of two a. Times B. To the exponent for and then we're going to subtract a log and again the same base in this case is D. Of the denominator of our division which is c. to the exponent nine. Alright so now we have this more expanded expression we have no more division inside of our log. What we do have in this first term is two things being multiplied together. So let's try to deal with that. Now we call that if we have log base B. Of X times Y. We can expand this and write this as log base B. Of X plus log base B. Of Y. Okay. So it's very similar to the rule for division. If we have multiplication inside of a log we can write this as the addition of two logs. Where the arguments of the logs that we're adding are the arguments of the inside of that multiplication. Those two things that are being multiplied together become the arguments of each of those separate logs and we keep the same base through. So if we apply this to the first term in our expression we're gonna have log base de of the first thing that's being multiplied. Which is the square root of two. A. And then we're gonna add log base de of the second thing being multiplied which is B. To the exponents for. And we still have our minus log base d. of c. to the exponents nine. All right now these last two terms K. This log based D of B. To the exponent four. And log based dfc to the exponent nine. Those are almost as expanded as possible. But we have these exponents. So we're going to deal with those. But let's deal with this log based D. Of the square root of two A. First. Because that's still a little bit messier than those other terms. Now recall that the square root can be written as an exponent. Okay so recall that A. To the exponents or the ending roots? Sorry the root of A. Can be written as A. To the exponent one over a. Okay. And so our square root here we can just write it as Log base de of to a. All to the exponent 1/2. And then we have our other terms log base d. of B to the four minus log base de of sea to the Not now we have exponents in each of these terms. So let's go ahead and deal with those all at once. Now recall your log rule for dealing with exponents. If we have log base B. Of X to the exponent N. We can write this as n log base B of X. So an exponents inside of the logarithms we can remove the exponents and it turns into a constant that multiplies the logarithms from the outside. So applying this to our expression in the first term we have an exponent of one half. We're going to bring that out in front and we get one half log base de of to A plus the second term. We have an exponent of four that four comes out in front. We get four log base D. Of B. And in this last term we get minus nine log based D. Of C. Alright so the last two terms those are fully expanded. We can't do anything else with those. But we have this first term inside of our log. We have to a okay now recall what we did when we had two things multiplied together inside of a log. And we use this rule above this log base B. Of X. Y. Is equal to log base B. Of X. Plus log base B. Of Y. We're going to apply that same rule again for this term here. So we're gonna have one half and I'm gonna put a big square bracket here. Okay That one half applies to this entire log. So when we break that log up into two things added together to separate logs added together, that constant one half has to apply to both of them. Okay. So it's a good idea if you put a big bracket here, that's a good reminder to yourself not to forget to apply that constant to both terms. So we get 1/2 log base de of the first thing that's multiplied inside which is going to be too. And then we're gonna add log base D. Of the second thing being multiplied. Which is a. Okay we close our bracket and then we have our final two terms that we're leaving alone. Four log base D. Of B minus nine log based D. Of C. All right, let's expand this. The last step we have to do is just to expand and distribute that one half to both of those terms. We're going to get one half Log Base d. of two plus one half blog based D. Of a Plus four Log Base D. of B minus nine log base D. Of C. And that is going to be our fully expanded expression. And if we go back up to our answer choices and we compare with what we've found, we see that we have answer choice A. We found that log based D. Of 2/2 plus log base B. Of a over two plus four log base B. Of B minus nine log based D. F. C. Is the expanded version of that original expression. Thanks everyone for watching. I hope this video helped see you in the next one.

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. logb ((√x y^3)/z^3)

Key Concepts

Properties of Logarithms

Logarithmic Expansion

Evaluating Logarithmic Expressions

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