Determine the solution set of the equation log to base eight of 𝑥 minus six plus log to base eight 𝑥 plus six equals log to base eight 64 in the set of real numbers.
To enable us to begin to solve this problem and actually solve the equation, all we actually want to do is apply a log law. And the log law that we’re gonna be using is that log to base 𝑎 of 𝑚𝑛, so that’s 𝑚 multiplied by 𝑛, is equal to log to base 𝑎 of 𝑚 plus log to base 𝑎 of 𝑛.
So therefore, if we look back at our equation, we can see, well, actually they all have the same base cause the eight is our base. It’s the same as our 𝑎 in our log law. So therefore, we can say that our log to base eight 𝑥 minus six plus log to base eight 𝑥 plus six is equal to log to base eight of 64 is equal to 𝑥 minus six multiplied by 𝑥 plus six is equal to 64. And that’s because if we look across to our log law, we can see that well the 𝑥 minus six is like our 𝑚. And the 𝑥 plus six is like our 𝑛. And if we actually multiply them together, that’s gonna be equal to our 64, which was our 𝑚𝑛.
And now we can solve this equation. And the first thing we need to do is expand the parentheses. So we get 𝑥 squared plus six 𝑥 minus six 𝑥 minus 36 is equal to 64. And then we can actually collect our terms here.
A key thing here as well to have a look at is the fact that these two parentheses are actually the difference of two squares. So what it means is that, actually, our 𝑥 terms are gonna cancel each other out. So we’re just gonna be left with our 𝑥 squared minus 36. And this is equal to 64.
Again, we say they cancel each other out. What that means is because if we have plus six 𝑥 and then minus six 𝑥, that’s just zero. Okay, so now our next step. Well, for our next step, we actually add 36. So therefore, we get 𝑥 squared is equal to 100. And then if we root each side, we get 𝑥 is equal to 10. So therefore, we can actually say that the solution set of the equation log to base eight of 𝑥 minus six plus log to base eight 𝑥 plus six is equal to log to base eight 64 is 10.