This expression is equal to the log of base b of 5. Great thinking if you got
this answer. Now, to start, I think it's best to think about this as two
logarithms of base b of 5. If I write this line all out. I see I have log base
b of 2 plus log base b of 5. Plus log base b of 5, minus log base b of 10. And
notice that there are two log base b's of 5 here. Now I'm ready to apply my
properties of logarithms. I know when I add two logarithms together that have
the same base, I can really just multiply these numbers together. So have log
base b of two times five. I'm going to do this one more time. I have one more
addition here, so I can multiply these number times these numbers. So we have
log base b of 2 times 5 times 5, minus log base b of 10. Now when we subtract
logarithms, we take this number and divide it by this number. So we'll have log
base b of 2 times 5 times 5 divided by 10. Now, you might be wondering why I
didn't go ahead and multiply these numbers. And it's because I knew that I
would have to simplify in the end. So I can see that 2 times 5 times equals 10,
and that reduces with another ten in the denominator. These simplify to one.
And this is how we get our final result of log base b of 5.
