Product Rule of Logarithms - Problem 3

So I’m actually going to go and prove the product rule of logarithms for you so you can actually see that it’s not some crazy mystery of a formula. So what I have is I have the product rule written up above and I have two claims, let m equal to log base b of x and n equal to log base b of y.

So what I want to do with these two is to first go and write them both into exponential form. So what we have here is b to the m is equal to x and then b to the n is equal to y. So what I want to do from here is actually multiply the two, multiply x and y together. So I want to look at x times y which is actually going to be b times m times b to the m time b to the n.

Remember when we are multiplying bases we can add our exponents, so what we actually end up with is xy is equal to b to the m plus n.

From here we take this and put it back into logarithmic form. This is exponential form something is equal to something else to a power, b is our base so that’s going to come around and we end up with log base b xy is equal to m plus n.

So we have our left side to be the same log base b of xy and now we also just have m, plus n. But we know what m and n are, we define them right here. This better place in this back in we end up with our formula log base b of xy is equal to log base b of x plus log base b of y.

So by just making this simple claim we are able to very easily prove the product rule of logarithms.