In Exercises 54–57, use properties of logarithms to condense each...

Question

In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. 1/2 ln x - ln y

Accepted Answer

Welcome back. I'm so glad you're here. We have got a log rhythmic expression here and it is A L N x minus one half. L N Y. We need to write it as a single log rhythmic expression. That means instead of expanding a single expression, we are going to condense this into one expression and we get to use properties of logarithms to do so. So let's look and see if we've got anything going on that reminds us of a property. Okay, well both of these L. N. S have a number hanging out in front of them. And that harkens back to the power rule. If you recall from previous lessons, The power rule says that when you have log B of M to a power, it is equivalent to sticking that power that P value in front of the log B of M. Those two sides are the same. So this time ours looks like the second one with the P number out in front but we can move that so it becomes this exponents for R. M. So we'll take our A and we'll have x rays to the A. And we'll take our one half and we'll have our Y raised to the one half. So let's write that, eh excuse me, Ln. Of X. To the A minus L. N. Of Y to the one half. So we've written that. Now any other properties that we can see at work here. Well, these two natural logs are being subtracted and that can help us recall the quotient rule, The quotient rule states that when we have a log B. Of two things being divided, A quotient M divided by N. That is equivalent to having log B. Of M minus log B. Of N. As long as those log bees are the same thing in this case they are. It's the natural log the L. N. And because those two sides are equal, we can go from the right to the left. We've got this subtraction happening here and we can just take our M. Which is our X. To the ninth and divided by R. N. Which is our Y. To the one half. So let's write that now. We've got the natural log L. N. Of A. And R. M. Term was X. To the ninth on top and our intern was Y to the one half on the bottom. And that's about as simplified as we can get though. We can rewrite this why to the one half we know that that's equivalent to the square root of why. So it's the same thing as the natural log L. N. Of X. To the ninth over the square root of why we look at our answer choices and that matches with answer choice B. Well done. We'll catch you on the next one

In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. 1/2 ln x - ln y

Key Concepts

Properties of Logarithms

Natural Logarithm

Condensing Logarithmic Expressions

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