Write the log expression as a single log.ln3xy+2ln2y−ln4x\ln\frac{3x}{y}+2\ln2y-\ln4xlny3x+2ln2y−ln4x

ln⁡(3y)\ln\left(3y\right)ln(3y)

Write the log expression as a single log. ln 3 x y + 2 ln 2 y − ln 4 x \ln\frac{3x}{y}+2\ln2y-\ln4x ln y 3 x + 2 ln 2 y − ln 4 x

In this problem, we're asked to write the logarithmic expression as a single log. And here I have the expression natural log of 3x divided by y plus 2 times the natural log of 2y 2y minus the natural log of 4x. Now here I wanna write this as a single log, so that tells me I'm going to be condensing these logs down into 1. So when we are condensing, we're always going to start with our power rule in order to get any of our multiplication up into that exponent. So the first thing I'm gonna do here is take the 2 in this 2 times the natural log of 2 y and bring it to the exponent. Now in doing that, I wanna make sure I am applying that exponent to both terms there, that 2 and that y, so this will become the natural log of 4 y squared because I squared both the 2 and the y. Now my other terms will stay the same. I still have the natural log of 3x+y and that's plus natural log of 4ysquared and then minus the natural log of 4x. Now that we've applied our power rule, we can go ahead and see what else we can do in order to condense this expression down. Now, I'm going to go through this expression left to right and just see what I can do here. Now, I first have this addition happening. And whenever I have addition, that tells me that I can go ahead and use my product rule because that addition will become multiplication inside of a single log. So I'm gonna go ahead and combine these 2 right here, and this will become the natural log of 3x, and then I'm multiplying it times that 4y squared. And then I still have that division from my first natural log, and it's over y. Now this natural log of 4 x is still just gonna come down here, natural log of 4 x. And I'm gonna go ahead and simplify this a little bit more because it can be simplified down. So canceling out this y here, I'm just left with that y on the top, and then 3 times 4 will give me 12. So this is really just the natural log of 12 x y, and then I have minus the natural log of 4 x. Now that I'm here, I see that I have this subtraction happening. And whenever I'm dealing with subtraction, that tells me that I can go ahead and turn this into division inside of a single log. Now since I only have 2 logs left, that means I'm gonna condense these down into 1, and I'll be done after this. So going ahead and condensing these down, I get natural log of 12 x y divided by 4 x. Now I can actually simplify this further because I see that I have an x on both the top and at the bottom. So these are simply going to cancel. And then 12 over 4 will give me 3. So this all simplifies down into just the natural log of 3 y. Now this is my final answer here. All of these 3 logs just condensed down into the natural log of 3 y. And we're done here. Let me know if you have any questions.