Use properties of logarithms to expand each logarithmic expres...

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\log_{b} (\frac{\sqrt{x} y^{3}}{z^{3}})$

Accepted Answer

Hey everyone welcome back in this problem. We are asked to expand the given log rhythmic expression and then try to evaluate by using the properties of logs. Okay? And the expression were given is log base de the square root of two A. Times B. To the four divided by C. Overnight. Okay so just rewriting that expression were given log based D. Of the Square Root of two A. Times B. To the four. All over C. To the nine. Okay. All right. The first thing we have here is we have division. Okay so the main operation at this level is division. Okay. We have all of the stuff on the left divided by C. Two. The nine. Well let's recall our law group for dealing with division. If we have division within our log log base B. Of X over Y. Say we can write this as log base B and x minus log base B. Of Y. So applying this to our expression. This allows us to write this whole thing as log base D. Of the numerator which is the square root of two A. Times B. To the four minus log base B. Of the denominator. C. To the exponent nine. Alright, so we've dealt with the division. Now there's a few things we can do here. We can deal with this exponents. Um You can go ahead and do that. But what we're gonna do first is we're gonna deal with this multiplication. Okay so we have the square root of 28 times B. To the four. Within the log let's go ahead and deal with that. So similarly to with division, recall the logro for multiplication, it tells us that if we have log base B of X times Y. You can write this as log base B of X plus log base B of Y. Okay so the base remains the same. The multiplication within the log becomes the addition of two separate logs. Okay so here we're gonna get log base D. On the square root of two. A plus log base D. Of B. To the four minus log base B. Of C. To the nine. Okay. Alright so now we have exponent here. We could use our exponent rule. We have an exponent here here we don't quite have an exponent. We have a route. So let's go ahead and deal with that first. Okay now recall if we have the end route of em we can write this as M to the one over end. Okay that allows us to write this route as an exponent and then we can use our log exponent rule to simplify. Okay so just rewriting that first term. Okay, the square root becomes to A. To the one half and we have the remaining terms log base D. Of B. To the four minus log base de whoops and sorry I'm messed up both of these appear. These should both be log base D. Okay so it should be the same base when we're separating the logs the same base all the way through. Okay so log base D. F. C. To the nine. Alright so now we have exponents on each of our terms. Let's go ahead and use our log exponent rule. Okay now we're called and I'm going to start writing the rules on the left hand side here just to have more space. We have log base be of some value X. A. To the end. Okay well we can write this as N log base B of X. Ok so the exponent N come out and multiplies to the entire log. So applying this to each of these terms here, our exponent is one half. So we're going to get one half Log Base D. of two times a plus four. The Experiment four comes out for log base d. of v. And finally minus nine log base D. Of C. And just be careful that you're keeping your signs. So we have the negative on that last term. Alright so this looks pretty good. This last term doesn't look like there's anything more. We can simplify or evaluate the same with this middle term, this first term. We see that we have multiplication again. Okay so we have a product two times A. So let's go ahead and use that product rule again this blue product rule that we used before. We'll go ahead and use that again on that first term. Okay so we have one half, we're gonna do big square brackets. Okay because the one half applies to the entire log. So when we're expanding, when we're splitting it into two separate logs with our product rule that one half coefficient is going to apply to both of those terms because we do big square bracket. To remind ourselves to apply that one half to both terms Log Base de of two plus log beastie six square bracket. The remaining term stay for log base D. B minus nine log based D. F. C. And final step, we're just gonna expand that one half, distribute that coefficient to each term. One half log base de of two plus one half log base di a plus four log base D. Of B minus nine log base D. C. Okay so nothing else can be um expanded. We just have a single term in each of these logs we can't evaluate anything. Okay so that's the solution there that is going to correspond with answer a log based D. F. 2/2 plus log base B. Of a over two plus four log based D. F B minus nine log based dfc. Thanks everyone for watching. I hope this video helped see you in the next one

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\log_{b} (\frac{\sqrt{x} y^{3}}{z^{3}})$

Key Concepts

Properties of Logarithms

Radicals and Exponents in Logarithms

Simplifying Logarithmic Expressions Without a Calculator

Watch next

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb(xy3z3)\log_{b}\left(\frac{\sqrt{x}y^3}{z^3}\right)

Key Concepts

Properties of Logarithms

Radicals and Exponents in Logarithms

Simplifying Logarithmic Expressions Without a Calculator

Watch next

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\log_{b} (\frac{\sqrt{x} y^{3}}{z^{3}})$