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 √(100x)

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 logarithms. Okay. And the expression were given is log base seven square root of Y. Okay so rewriting that log base seven of the square root of 49 Y. We see we have our route here and we know that we have a logarithms rule that allows us to deal with exponents. Okay now let's recall if we have roots, we can write those as exponents. So let's go ahead and do that. So if we have the end the root of M. We can write this as M. To the exponent. One over N. Okay so applying this to our expression, the square root becomes and exponents of one half. So we get log base 7 49 Y. To the one half. Alright now we have an exponent and we can go ahead and use the log rhythm rule tells us what to do with an exponent. If we have log base B some value X. To the exponent end we can bring the exponents and down and in front and right end log base B of X. The exponent comes out and multiplies in front. In our expression we have an exponent of one half. Within the log it's gonna come out in front we're gonna get one half log base seven of 49 Y. Alright now that we have that. Well let's take a look inside our log again. We're trying to expand. So we're trying to look for anything that we might be able to expand here, we have a multiplication 49 times Y. How do we deal with multiplication within our log? We have a rule. Let's recall that says 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 can take the multiplication within the log and turn it into an addition of two logs with the same base. That's gonna give us one half big square bracket here. The one half applies to this whole log. And so when we break that log into two different terms, the one half applies to both terms. Okay so put a big square bracket here. So you don't forget to distribute that constant one half to both terms. Log base seven of 49 Plus Log Base 7 & Why? Okay, That looks good on the second term here. The Log Base seven of Why? There's nothing else we can expand there. But let's try to deal with this. Okay? We have log base seven of 49. Let's see if we can evaluate that and get a number out of that. Well let's see. We know Let's do a little aside, we have log base seven of 49. We know that if we have something like this equals to why we have a rule that says log base B of X equals Y. That we can write B to the exponent Y equals X. Okay, this allows us to convert from logarithmic expressions to exponential expressions. Okay, if we go ahead and do this and apply this to our problem. The bases seven? We're gonna get seven. We've left. Why the same in both? Okay. So he said seven to the exponent y Is going to be equal to whatever's inside the log. And in this case it's 49 case we have seven to the Y equals 49. Well, that tells us that seven to the y. We know 49 is seven squared seven squared. Okay? Because these are equal, the bases are the same. The exponents must be equal. So we're gonna get y equals two. Okay? So if we evaluate log base seven of 49 we're going to get to Okay, So we can substitute that in here and we're just gonna have to Plus Log Base seven of What? All right. Okay. Now the last thing we want to do is we're just going to distribute that constant term of one half to both terms. Okay, one half times two is just gonna be one plus one half log base seven. Why? There's nothing more to expand here? There's nothing more to evaluate. So this is gonna be our answer. We're gonna have one plus one half log base seven of y. That's going to be answer A Thanks everyone for watching. See you in the next video

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 √(100x)

Key Concepts

Properties of Logarithms

Radicals and Exponents

Evaluating Logarithmic Expressions

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