We will need to use the first two rules in the table to combine the logs, but those rules donβt work if there are numbers out front. So, we need to use the third rule in the table to deal with those first. We can bring the numbers inside the logs, but they have to go in the exponent.
Notice that in the last one we had to put parentheses around the \(x+1\text{.}\) The reason is that when we bring the exponent inside, it has to be an exponent on the entire inside. Without the parentheses, the \(\frac{1}{2}\) power would only be on the 1, completely missing the \(x\text{.}\)
Now we are ready to combine the logs into one. Since the log with \(y^3\) is being subtracted, our second rule tells us that the \(y^3\) will go in the denominator of our fraction.
Notice that when we brought the exponents inside, the log with the \(y\) only brought the 3 inside, but left the negative. If you prefer, you can bring the negative inside, too. Youβll just have some negative exponents to deal with, but because those move things to the denominator, youβll end up with the same answer.
We will need to use the first two rules in the table to combine the logs, but those rules donβt work if there are numbers out front. So, we need to use the third rule in the table to deal with those first. We can bring the numbers inside the logs, but they have to go in the exponent.
Now we are ready to combine the logs into one. Since the log with \(y^{\frac{3}{2}}\) and the \(z^7\) are being subtracted, our second rule tells us that those will go in the denominator of our fraction.