Often times we are given information about certain logarithms and asked to express another log in terms of that information. So in this case we have information about log base 5 of 2 and log base 5 of 3 and we're trying to figure out how to express log base 5 of 15 over 2 in terms of those two things.
Typically what we're going to have to do is just break down this log base 5 of 15 over 2 and the first thing I see is that I have a division inside my log so I can break that down as subtraction. So we end up with log base 5 of 15 minus log base 5 of 2. That's easy enough, so we actually have information about log base 5 of 2 so I can leave this statement alone.
The next thing we want to work on is the log base 5 of 15. And the trick is that I have information about log base 5 of 3, so I want to break down the 15 into 3 times something, so this becomes just log base 5 of 5 times 3 which when we're multiplying inside of our log we break down into addition. So this turns into log base 5 of 5 plus log base 5 of 3 and we still have this minus log base 5 of 2 at the end.
So we know that this is a 'j', we know that this is a 'k' we still have to evaluate log base 5 of 5. The cool thing is that we actually know what this is, 5 to what power is equal to 5? This is just going to be 1, so what we end up with then is equals to 1 plus k minus j.
So using our properties of logs, we are able to break this down to these components and just throwing in the fact that we know also what multiples of the base are, we're able to solve this out completely.