In Exercises 125–128, determine whether each statement is true or...

Question

In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

logb (x^3 + y^3) = 3 logb x + 3 logb y

Accepted Answer

Hey everyone in this problem we are asked to verify if the given equation is true and if the statement is false, were asked to make the necessary changes to amend the equation in order to make it true. Okay. And the statement we're given is log base end of a cube plus B to the four is equal to three. Log base 10 of a plus four log base 10 of B. Now, if we start with the left hand side of our equation, log base end of a cubed plus B to the four and we try to expand this to get it to look like the right hand side, you'll see that we don't have any rules that deal with addition inside of a log in order to be able to expand this. So we don't have a way of expanding this to look like the right hand side. That kind of gives us an impression that maybe this statement is false. But let's start with the right hand side instead, see if we can manipulate it and simplify it. To get it to look like the left hand side in order to see for sure whether the statement is true or false. So the right hand side, we have three log base 10 of a plus four log base B. We want to try to simplify this and combine it into a single log rhythm. Now the first thing we want to do is deal with these constants out in front. Okay, so we have a constant of three and four being multiplied by our logs, let's recall our rule for dealing with a constant out front of a log being multiplied. So if we have em log base B of X we can write this as log base B of X to the exponent M. Okay so the constant being multiplied outside of our log can come into the brackets and become an exponents on the argument of our log. So if we apply this to our problem, our expression the constant of three is going to go inside of the log and become an exponents on our argument. A. So we get log base N. Of a cute And same for the second term. The constant of four is going to go inside of the brackets and become an exponents on the argument be. So we get log base end of B to the exponent four. So now we've dealt with those constants, we have two logarithms being added together. Let's recall our log rule for dealing with two logarithms being added together. If we have log base B of X and we add log base B of Y we can write this as log base B of X times Y. So the addition of two separate logarithms with the same base, we can turn into a log rhythm with that same base where we multiply the two arguments together inside of the logarithms. So we take the argument of the first, log X. The argument of the second log Y. And we multiply them together inside of a single log. Alright so applying that to this problem we're gonna have log base end. Okay we keep that same base, we have the same base for each of these being added together and then we're gonna multiply the two arguments. So we have a cube from the first log rhythm and we have B to the four from the second long rhythm. So we get log base end of a cubed times B. To the four. And what you'll notice here is that this does not equal log base end of a cubed plus before. Okay within that log we can't change a cubed times before to be a cubed plus before those are not equal. Okay so that tells us that our original original statement is indeed false. So we're looking at answer choice B C R D. Because we know our statement is false. And what we just found is that three log base 10 of a plus four log base B is equal to log based end of a cute B. To the four. And that's the statement given in B. And so the answer here is going to be be our original statement log base end of a cubed plus B. To the four equals three log. Base end of a plus four log base 10 of B is false. And the correct statement is instead log base end of a cubed times B to the four is equal to three log base 10 of a plus four log base B. Thanks everyone for watching. I hope this video helped see you in the next one.

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In Exercises 125–128, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. logb (x^3 + y^3) = 3 logb x + 3 logb y