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.

log4 (√x/64)

Accepted Answer

everyone. Welcome back in this problem we are asked to expand the given logarithms expression and try to evaluate it by using the properties of logs in the log expression were given is log base seven of the third root of Y divided by 343. So log base seven of the third root cube root of Y Divided by 343. Alright so the first thing we want to deal with is this division. Okay so let's recall our log property that deals with division. If we have log base B of X divided by Y. You can write this as log base B of x minus log base B. And why? The division within the log becomes the subtraction of two logs applying this to our expression. We get log base seven of the numerator cuba of y - Log Base seven of the Denominator 343. Okay great. No on the left hand side we have a route and we know we can wright roots in terms of exponents. So let's recall the rule that allows us to do that. If we have the end route of them, this is equivalent to writing em to the exponent one over end. Okay so let's apply that rule here in the second term. Okay we have a base of seven in our log we know if we could write log base seven of seven That we would have won. Okay, log base seven of seven is just going to be one. Okay so let's see if we can write 343 With a base of seven. Alright so the first term we're going to get log base seven again rewriting this cube root in terms of exponents that's gonna give us y to the one third In. Rewriting 343 in Bay seven. Well we can write that as seven. Cute. Okay so seven times seven times seven is going to get 343. So we have seven cube there. Alright now both of our terms have exponents. Okay. And the first one we have exponent 1/3 and the second one we have exponents three. What is our log rule for dealing with experiments? What we call? If we have a log base B of some quantity X. To the experiment and this can be rewritten as n log base B of X. Ok. So the experiment comes down outside of a log and multiplies the entire log. Okay, so for our expression we have exponent of one third in the first term. So that's going to turn into one third Log Day seven of Why? The second term we have exponents of three. That's going to become three log base seven of seven. All right, great. All that first term looks simplified or expanded as much as we can. We can evaluate any further. Okay, this second term they'll recall log base seven of seven I said is going to be equal to one. Ok. So log base B of B. No matter what B is is just one. Alright, so we can rewrite the entire expression then as 1/3 log base seven of Y -3. And that's as simple as we can get it. And so our solution is going to be answer C one third log base seven of y minus three. 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. log4 (√x/64)

Key Concepts

Properties of Logarithms

Change of Base Formula

Evaluating Logarithmic Expressions

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