 In this video, we provide the solution to question number three for the practice final exam for math 1050. We have to expand the logarithm, well the common log of x cubed times the square root of x plus 1, all over x minus 2 quantity squared. So first of all, this is a fraction inside of the logarithm. So using the second law of logarithms, this would expand to be the log of x cubed times the square root of x plus 1 minus the log of x minus 2 quantity squared, like so. In the first logarithm, there is a product going on here. You have x cubed times the square root of x. So we're going to break that up with regard to the product there. So that gives us the log of x cubed plus the log of the square root of x plus 1 minus the log of x minus 2 squared. And so then with all the remaining logarithms, there are exponents that can be pulled out. The x cubed, the exponent comes out as a 3. For the square root, it will come out as a 1 half. And then for the minus log of x minus 2 squared, that 2 would come out as a coefficient. So the correct answer we're looking for would then be 3 log of x plus 1 half log of x plus 1 minus 2 log of x minus 2. For which we see that's exactly choice a, just like we said a moment ago, 3 log of x plus 1 half log of x plus 1 and minus 2 log of x minus 2. So using the three laws of logarithms, we're able to correctly expand this logarithmic expression. Thank you.