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 (xy)^5 = (logb x + logb y)^5

Accepted Answer

Hey everyone welcome back in this problem. We are asked to verify if the given equation is true. Okay and then make the necessary changes to amend the equation if it isn't true. Okay so that we can write a true statement. Now the statement we're given is log base B of M N cubed is equal to log base B of M. Plus log base B of N. All cute. Okay so let's start with the left hand side. We have log base B. Of M N. Cute. Mhm. Now let's recall if we have an exponents inside of our log, let's recall the rule that allows us to deal with that. We have a log property that states if we have log base B. Of some A. To the exponent N. The exponent can come out of the log and multiply in front. Okay so we can write this as n log base B of A. Okay, so applying this to our statement or our expression here, we're gonna have the exponent three come down and multiply in front. Okay so we get three log base B. Of the inside em. Alright so we've gotten this to this stage but we see in the statement that we're trying to compare it to, we have the M. In the end, split into separate logs. Alright so let's go ahead and let's think about what properties we have to deal with terms that are multiplied inside of a lock and we can write that log base B of X times Y. Is equal to log base B of X. Let's log base B of why? Okay this is one of our properties. So we take the multiplication inside of a log and we can turn it into addition of two separate logs all with the same base. Okay applying this to our expression we get three times and we're gonna put big square brackets so we remember that that three needs to apply to this entire log which is going to be both of the terms. Okay so we have three times log base B of the first thing em plus log base B of the second thing. And alright now we have a simple log single base single thing inside. So there's nothing else we can do to simplify this. And what we'll see is that the statement we were given is false. Okay log base B of M N cube does not equal log base B of M plus log base B of M cubed. But it does equal what we just found three log base B of M plus log base B of N. Okay, so the answer is going to be be, the original statement that we were given is false. And the true statement is that statement we found afterwards. Okay, that's it for this problem. I hope that helped see you in the next one

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 (xy)^5 = (logb x + logb y)^5

Key Concepts

Properties of Logarithms

Exponentiation in Logarithmic Functions

True and False Statements in Algebra

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