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Combine the given expression into a single logarithm.$$\frac{1}{4}[3 \ln (x+3)+\ln (x+1)-2 \ln (x+5)]$$

$$\ln \frac{\sqrt[4]{(x+3)^{3}} \sqrt[4]{x+1}}{\sqrt{x+5}}$$

Algebra

Chapter 4

Exponential and Logarithmic Functions

Section 6

Properties of Logarithmic Functions

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e think this problem probably looks scarier than anything else because of the brackets in there. Um, but if you just do this piece by piece and it's gonna make perfect sense of X Plus three and then plus natural log of X plus one and minus to natural log of X plus five e Just double check that I copied everything in their correctly. Uh, because what I would do first is ignore this 1/4. I'll worry about that. At the end, I would move the exponents Ah ah, that air inside the brackets. So ignore that 1/4 for a second. So we're looking at still bracket, though. Natural log of X plus three cubed plus natural log of X plus one. And I'm not messing that yet. And a minus natural log of X plus five squared. So from here may what I would do is think about okay, if we're adding that we need to multiply with a single logs we're looking at natural log of X plus three cubed, uh, Times X plus one. And then when you are subtracting that is a division eso the X plus five squared goes into the denominator Now you still have this 1/4 eso that's written in front because I could probably drop the brackets now at this point and for you to understand that that 1/4 is going to become an exponents to the entire term. So, like dropping the use brackets and sticking brackets inside. And the only other thing to mention then is that everything is taken to that 1/4 power. So I would write the answer this way is natural. Log as the 1/4 power is the fourth root of everything. Now, you could individually do the fourth root of each piece, but this is acceptable as well. So I'm gonna circle my answer and move on. Uh huh. Okay.

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