Hi Jenny from Chicago...Michael from Itasca here.
So, first, I think we need to make sure we understand the problem. Those numbers after the logs are their bases, right?
x = log25 and y = log109
And we're looking for log23. That's our target.
Here are some hints.
1. I notice that y is the log of 9, and our target has a log of 3 in it. Can't we write y as some form of a log of 3, using one of the properties of logs.
2. If we can do that, then we still have a problem, because y is written in base 10, and our target is in base 2. But can't we fix that? What would our target be in logs of base 10? Can you convert the base?
3. If we can convert the base of our target, then we should end up with a log3 somewhere, which is related to y. If you figured out Step 1, that is. :)
That takes care of the log3 part...but when we convert the base of our target, we're going to end up with a log2 to deal with, too. In fact, you'll have 1/log2 if you did it right.
4. Don't we know something about 1/logba? It's not one of the top four properties of logs, but it's right behind the most common ones. If we can turn the log102 into a log with base 2, then it's going to start looking a lot like
x, because x is a log of base 2, also.
6. In fact, x = log25. If we had that exact log, then we'd have x, and we'd be done. But that never happens, does it. :) How about if we had some sort of variation of that, like log2(5*n). Couldn't we use another property of
logs to split out the part that looks like x?
7. And if we could do that, wouldn't it be convenient if "n" were a number that we could deal with, like a power of 2? Because then, we'd be able to figure out what the log2n is, too, using yet another property of logs.
Between those hints, and the fact that you know the answer, can you piece together the puzzle?
Hope this helps more than hurts,