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.

log6 (36/(√(x+1))

Accepted Answer

everyone. Welcome back in this video. We're asked to expand the given logarithmic expression and then try to evaluate it using the properties of logs. Okay the expression we're given is log base nine of 81 divided by the square root of x minus nine. Okay so rewriting what we're given log base nine of 81 divided by the square root of x minus nine. Alright so the first thing we need to deal with here is the division. Okay, can't deal with the square root until it is out of that denominator. Okay so we want to recall our log rule for division if we have log base B of X divided by Y. Okay well we can write this as log base B of X minus log base B of Y here. So the division within the log becomes a subtraction of two log. Okay so applying this to our expression we're going to have a blog base nine of 81. Okay minus log base nine of the denominator squared of x minus nine. Alright great. No the first term log base nine of 81. There's a few ways that we could try to evaluate this. Okay um we could try to evaluate this by using a rule that allows us to convert this log into an exponential equation. Okay and then solve the exponential equation. Let's try to do it here with exponents. Okay so let's try to write both of these with exponents. So let's recall our rule for writing roots in terms of exponents. If we have the end route of em we know we can write this as M. Two. The one over N. Alright so what about this first? We have a base of nine. Okay we know that log base B of B is equal to one. Ok So if we can write log base nine of nine we know that's equal to one. So let's try to write 81. In terms of base nine, we know that's nine squared. So 81 times nine squared. Okay minus log phase nine Of X -9 to the one half. Okay. Using this rule on the right hand side where we convert the root into an exponential. All right now both terms have exponents. Let's recall our log rule for dealing with exponents. We have log base B of some value X. To the exponent N. We can write this as n log base B of X. Okay so the experiment can come out and multiply the entire log. So on our first term we have expanded to so we end up with two log base nine of nine K. And recall log base nine of nine. Well that's just gonna be one. Okay so moving to the next line that allows us to write that as one and we've completely eliminated the log and evaluated that expression for our second term. We have an exponent of one half. So we're gonna get one half Log Base nine of X -9. Alright so simplifying the first expression like I said okay this log base nine of nine is one. So all we're gonna be left with is 2 -1/ log base nine of X -9. And you might be inclined here you see this log-based nine of X -9. You might be thinking oh I can apply the subtraction rule. Turn it into a division. No okay when we're using that rule recall this rule appear we need to have division within the log. Okay to separate it into a subtraction of two logs we don't have a rule that tells us what to do or how to simplify if we have subtraction in the log. Okay so this rule, this rule here does not apply to this term x minus nine and we actually don't have a rule that tells us how to expand or simplify that. Okay so that's going to be left alone. We can't expand or evaluate it further. And so our solution is going to be two minus one half log base nine of x minus nine. Okay and that's going to correspond to answer D. Thanks everyone for watching. I hope this video helped

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. log6 (36/(√(x+1))

Key Concepts

Properties of Logarithms

Logarithmic Expansion

Evaluating Logarithmic Expressions

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