 An important skill for working with logarithms is the ability to rewrite logarithmic expressions. Using the rules of logarithms, we were able to expand a single logarithmic expression into a sum, difference, or product-involving logarithms. It will also be helpful to go backwards to rewrite a sum, difference, or product-involving logarithms into a single expression. And as usual, it helps to identify what type of expression we have. So here, log2 of x plus 10 minus log2 of x plus 1, this expression is a difference of logs. And if we consult our rules of logs, we see that a difference of logs shows up when we consider a quotient. So this difference of logs becomes the log of the quotient x plus 10 divided by x plus 1. How about something like this? So remember, the type of expression is determined by the last thing that you do. And this is minus one-third times log of x. And that means that times is the last thing we do. This is not actually a log, at least not yet. Now, while we could handle this coefficient of minus one-third, it's sometimes helpful to rewrite negative numbers as differences. So instead of minus one-third log x, I can write this as zero minus one-third log x. So what does that do for us? Well, we might remember that zero is the same as log of one. And now I have a difference which suggests I'm working with a quotient except this isn't yet a difference of logs. It's log minus one-third of a log, which means we have to do something with that coefficient of one-third. Well, let's take a closer look at that. That's one-third times log, and that sounds a lot like number times log, and that comes from the power rule for logs. And so this one-third log x, I can rewrite this as log of x to the one-third, or I can also write that as cube root of x. Now we have a difference of logs, so now we can use the quotient rule. And then we have something like this, and the most important thing to recognize here is that this is not a sum of logs. What we're actually adding is three times a log plus two times a log. Which means we have to do something with this three times and the two times. We can use the power rule and rewrite our two expressions. And now we have log of something plus log of something, and that sounds like the product rule, which gives us our final expression.