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^2 y)/z^2)

Accepted Answer

everyone in this problem. We are asked to expand the given log rhythmic expression and then try to evaluate it by using our log rules. And the expression were given is log base X. Of a cube B squared. All divided by C. Two the seven. Okay so just rewriting what we're given log base X of a cubed B squared divided by C. Two. The seven. Okay. Alright so looking at this, we see that the main operation that we need to deal with first is the division. Okay so we have everything in the numerator divided by everything in the denominator. So let's recall our log rule for dealing with division. If we have log base B of X over Y we can write this as log base B of X minus log base B of Y. Okay so the base remains the same. The division within the log becomes a subtraction of two different logs. So applying that to our expression we're going to have log base X. Of the numerator, A Q B squared minus log base X of the denominator. C. To the seven. So log base X A Q B square minus log base X of C to the side. Okay now we have an exponent in this last term. We have multiplication in this first term. Okay so let's go ahead and deal with the multiplication first. Either way it's fine but we're going to deal with this multiplication first. And what we see is we have a Q B squared. So similarly to the rule for division we have a rule for multiplication within logs. Okay so let's recall log base B of X times Y. Can be written as log base B of X plus log base B of Y. Okay so we keep the same base the multiplication within the log becomes the addition of two logs. Okay so applying this to our rule or to our expression, sorry we're gonna have log base X of a cubed plus log base X of B squared. Okay. We split that multiplication up and finally our minus log base X. Of C to the seventh. Okay. And just be careful that you're keeping that sign all the way through and you're not mixing up any signs. Alright great. So now we've separated our three terms. We have eight ace here bees here C. Is here. And what we see is that each of these terms has exponent. Okay well let's use our log exponent rule. Does the log exponent rule say? And we're going to just write it over here on the left hand side so we have some more room. Well if we have log base B. Of some quantity A two exponent N. Let's let's not use A. Just because we have A. And R. Problem. Let's use why to experiment and Okay well we can rewrite this as N times log base B of Y. So the exponent end comes out in front and multiplies to the entire lock. Okay so applying this rule here, this blue rule to this following step, we're gonna apply it to each term. We're going to end up with three log base X. Of a. Okay, so that exponent of three has come down and multiplied Plus two Log Base X of B - Log Base X of C. All right, So all the exponents have come down and multiplied. And that expression there, we can't expand anything else. We can't evaluate anything else. So that is going to be our solution. So we get three log base X of a plus two log base X of B minus seven log base X of C. That's going to be answer B. Thanks everyone for watching. Hope to see you in the next video.

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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^2 y)/z^2)

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