Hey, everyone. We just learned all of the rules needed to expand log expressions, but two things are going to happen. We're going to be asked to expand much more complicated log expressions, like, say, log2(3xy2). And we're also going to be asked to do the exact opposite of expanding log expressions and actually condense expressions with multiple logs down to a single log. Now, you don't have to worry because we're going to take these rules that we already know, and I'm going to walk you through how to use multiple of these rules and how to use them in their reverse direction because each of these properties can be applied in both directions depending on what your goal is, whether taking a single log to multiple logs or condensing multiple logs back down into a single log.
Now you'll be able to expand and condense any log expression that gets thrown your way, so let's go ahead and get started. Now we're going to just jump right into an example here. And in our first example, we see that we have log2(3xy2), and we want to go ahead and expand this log expression as much as possible. So the first thing I notice here is that I have 3 different things being multiplied together. This is actually 3 times x times y2.
Now, since this is multiplication, that clues me in that I need to go ahead and use the product rule because I know that the product rule takes multiplication and turns it into the addition of multiple logs. So let's go ahead and expand this log out into multiple logs using that product rule. So this is log2(3) plus log2(x) plus log2(y2). Now I have these multiple logs, but I want to go ahead and walk through each of these single logs to make sure that I can't expand them any further. So I have log2(3).
I can't expand that anymore, so that's going to remain as a log2(3). And then log2(x), I also cannot expand that anymore, so that will remain the same as well. And then finally, we come to our last term, log2(y2). Now, looking at this log, I know that I have this exponent here. And whenever I have an exponent, that tells me that I can go ahead and use the power rule because the power rule tells me that I can take that exponent and pull it to the front of my log to expand it.
So this 2, I can go ahead and pull to the front of my log, and this last term becomes 2 times log2(y). Now none of these terms can be expanded anymore, so this is my final answer, log2(3) plus log2(x) plus 2 times log2(y). We've taken this one log and expanded it out as much as we can. Now let's look at condensing a log expression. Now, we do have to consider a couple of additional things whenever we're condensing logs.
And the first is that we always want to make sure that the base has to be the same. So here we have 2 times the natural log of x minus the natural log of x plus 2. So both of these are natural logs. They have that same base of e. If this was, say, the natural log and log base 2, I couldn't condense those logs together because they have different bases.
We'll always apply the power rule first to get the correct answer. So we have 2 times the natural log of x minus the natural log of x plus 2. I'm going to look at these terms and see how I can apply that power rule. So I have 2 times the natural log of x. I can take that thing that's multiplying it and make it into the exponent. So this becomes the natural log of x squared.
That second term, I can't apply the power rule, so it's going to remain the same for now, the natural log of x plus 2. How else can we condense this expression? They're being subtracted, so I should go ahead and use the quotient rule. I can take subtraction and turn it into the division of a single log expression. So this becomes the ln(x2×(x+2)). Now I have taken these multiple logs and condensed them down into a single log, so this is my final answer, the natural log of x2x+2. Now that we have seen how to expand and condense log expressions, let's go ahead and get some more practice. Thanks for watching, and let me know if you have questions.