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Quotient 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!

Proving the quotient rule of logarithms. For this little episode what I’m going to do is actually take some time and show you where this formula comes from.

So in doing that the one we have to do is assign two different logs; m is equal to log base b of, x n is equal to log base b of y. First step what I want to do is take both of these equations that are in logarithmic form and put them into exponential form. Bring the bs around, this will give us x is equal to b to the m and y is equal to b to the n. What I then want to look at is the x over y, which is the same thing as b to the m over b, try that again, b to the n. Knowing our properties of exponents when we’re dividing bases are the same we can rewrite this as subtraction. This is the same thing as b to the m minus n.

Rewriting this we now have x over y is equal to the b to the m minus n. This is in exponential form. So what we can do is put is put it back into logarithmic form giving us log base b, x over y is equal to m minus n. We know what m and n are; they’re what we defined to be right here. By simply rewriting this, replacing the m and the n we end up with log base b, x over y is equal to log base b x minus log base b, y.

By just doing this simple part rules of exponents and making an assumption about m and n we we’re able to prove the quotient rule for logarithms.

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