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. log 3 - 3 log x

Accepted Answer

Welcome back. I'm so glad you're here. We've got this log rhythmic expression which is one third, log X. Plus two Log Y. And we're going to combine it to a single log rhythmic expression. So instead of expanding it, we're combining it into one log rhythmic expression. And we're still going to use those properties of logarithms. And now let's look and see if there's anything going on that reminds us of one of the properties. Well both of these logs have a number floating out in front of them and that reminds us of the power rule. If you recall from previous lessons, the power rule states that when you have log B. Of M to a power it is the same as that P value out in front of log B. Of M. And in this case because we are condensing rather than expanding, we're looking at the right hand side for what we already have, that P. Value out in front. But we know we can move it so that it becomes the exponents of R. M. Value. So the first one can be rewritten that one third is going to become the exponents of our X. That too is going to become the exponents of Ry will rewrite it log and this is just a log that B value is our base 10. So we don't it's understood, we don't have to write it. So log X to the one third plus there are two moves over. So it's log why squared great any other properties going on here while we see these two logs are being added that can remind us of the product rule. The product rule states that when we have log B of M times N, it is equal to log B. Of M plus log B of N. When those two log bees are equivalent. And in this case they are they're both log base 10. This time we're looking at the right hand side because it's already expanded, those two logs are being added. They have that same log base. And so we can combine them in multiplication. So we can rewrite this as log and then R. M, which is X to the third and X to the one third and Y squared are now being multiplied by each other. So log X to the one third times Y squared. Those are M. And end terms now being multiplied. And that is as much as we can do. Following the properties of logarithms. We look at our answer choices and that matches with answer choice. B. While done, we'll catch you on the next one.

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In Exercises 54–57, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. log 3 - 3 log x