In Exercises 50–53, use properties of logarithms to expand each l...

Question

In Exercises 50–53, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln ∛(x/e)

Accepted Answer

Welcome back. I'm so glad you're here. We have got this logarithms here Ln X E raised to the X squared and we are going to expand it. And so let's look and see what's happening here with our log rhythmic expression and see if that helps us recall any of our properties of logarithms. Okay, We've got two terms here that are being multiplied. That can help us recall the product rule. If you recall from previous lessons, the product rule states that when we have log B of M and meaning to terms being multiplied together, it equals log B of M plus log B of N. So we get to expand them by adding them. So let's write that now. We've got Ln the natural log of X. The first charm here plus the natural log L. N. Of E to the X squared. Has that second term here. Alright, we've got one expansion done. And now is there anything else we can do that first term is kind of done. The second term. We've got Ln of E to the X squared. Well that Ln of E to the X squared, there's a power going on here that is raised to a power. And that can remind us of the power rule. Recall from previous lessons that the power rule states that when we have log B of M raise to a power, it equals that power value out front of log B of M. So in this case the powers the X squared and we can just move that in front of our natural log of E. So let's rewrite that that way. We've got L. N. Of X plus X squared, L N E. And now another lot of another property that we've got at work here is this L N E. Well, L n E. We recall from previous lessons that L n E just equals one and so that X squared is just being multiplied by one. Or in other words we've just got Ln X plus X squared. We can rewrite that with the X squared first X squared plus natural log of X. And that's as simplified as it's going to get. So we look at our answer choices and that matches with answer choice. C. Well done. We'll catch you on the next one.

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