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Product Rule of Logarithms - Problem 3

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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.

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