Solution

\[ \solve{ 2\log(x-5)-\log(x)+\frac{1}{3}\log(y+1)-3\log(y)&=& \log((x-5)^2-\log(x)+\log((y+1)^\frac{1}{3})-\log(y^3)\\ &=&\log\left(\frac{(x-5)^2}{x}\right)+\log((y+1)^\frac{1}{3})-\log(y^3)\\ &=&\log\left(\frac{(x-5)^2(y+1)^\frac{1}{3}}{x}\right)-\log(y^3)\\ &=&\log\left(\frac{(x-5)^2(y+1)^\frac{1}{3}}{xy^3}\right)- } \] As before, we process the Power Rule first before applying the Sum/Difference rules. When deciding the correct order, we read from left to right and do each subtraction or addition as they come. Finally, you can do a mental check that any logarithm in the initial expression that was negative should contribute to the denominator only, and positive logarithms contribute to the numerator only.