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. log4 (√x/64)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this log rhythm log base two of 32 X. To the fifth. And we are going to expand it as much as we possibly can. So let's look and see what we've got going on here. That helps remind us of some of the properties of logarithms. Well this 32 is being multiplied by the X. To the fifth. When two things are being multiplied in the log. That can remind us of the product rule. If you recall from previous lessons, the product rule states that when we have log B. Of two things being multiplied M times N. That equals log base B. Of M. Plus log base B. Of N. And so our m. term is that 32. And that end term Is our X. to the 5th. So we can now write that expanded. We've got log two of RM term 32 plus log base two of X to the fifth. Cool. We've expanded it with the product rule. What else is going on here? Well, we look at log base two of 32 we want to think about 32. What's another way to express? 32. 32 is the same thing As 2 to the 5th. So we can rewrite that now as log two of 2 to the 5th plus log two of X. To the fifth. And now what are we noticing? Well they're both raised to a Power, they're both raised to the fifth. And that can help remind us of the power rule we recall from previous lessons that when we have log B. Of M. To a power. It's the same thing as putting that power value in front. P log B. Of in. So our p values here are five and we're just going to scoot those in front. And so we've got five log two of two plus five Log two of X. What else do we know here? Well Log two of 2. We know From previous lessons that that is equivalent to one so that five is just being multiplied by one. So that's just five plus five Log 2 of X. Did those two have anything in common? Yeah they both got a five. So we could factor a five out in front. That first term is now 15 divided by five is one plus that second term. There's a one there. We don't have to write it. We can just write log two of X. We look at our answer choices that's as simplified as it's going to get and that's going to match with answer choice. C. Well done. We'll catch you on the next one.

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