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.log ∛(x/y)

Accepted Answer

Hey everyone welcome back in this problem. We are asked to expand the given logarithmic expression and then try to evaluate it by using the properties of logs and the expression were given is log of the square root of A. Over B. Okay so we have log of the square root of A. Over B. Okay now the first thing we want to do because we want to take this square root and instead write it as an exponent. Okay we know we have log rules to deal with exponents. So let's go ahead and write that as an exponent. So if we have the end route of some quantity M. Recall that we can write this as M. to the exponent one over N. So applying this to our expression we can write log Of a. Over b. to the exponent 1/2. So we've just rewritten our square root to be to the exponent 1/2. Now let's recall our log rules for dealing with exponents. Okay if we have exponent within our log log rule that tells us if we have log base B of X to the end, we can bring that exponent out of the log rhythm and multiply it at the front. Okay so we're gonna have n log B of X. Okay so applying this rule with our expression, the exponent of one half comes out and in front and we're gonna multiply. So we have one half times log of A. Over B. All right now in our log we see that we have a division. Okay well let's recall what do we do when we have division inside of our log? Well if we're dividing we have log base B. Of X over Y. We know we can write this as log base B of x minus log base B. Of Y. Okay. So we keep the same base of our log. We have the numerator comes out by itself and then we're subtracting the log of the denominator. So in our case we have one half we're gonna put a big square bracket here. Okay this is important. It can be easy to mix up this. So the one half needs to apply to this whole logarithms which means it needs to apply to both terms that are going to be in our subtraction. Okay. So put a big square bracket here so that you remember to distribute that one half to both terms. Okay so then we're gonna have log of a minus log B. Okay? And then just expanding and distributing that one half. Like we said we got one half log of a minus one half log of B. Okay. There's nothing more we can do. We just have a simple a simple be within our log. There's nothing more we can expand there. Okay so this is going to be our answer. So we're gonna have answer. A one half log a minus one half log B. I hope this video helped. Thanks everyone for watching

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.log ∛(x/y)

Key Concepts

Logarithmic Properties

Change of Base Formula

Radicals and Exponents

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