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Quotient Rule of Logarithms - Concept

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!

When evaluating logarithms the logarithmic rules, such as the quotient rule of logarithms, can be useful for rewriting logarithmic terms. The quotient rule of logarithms allows us to separate parts of a quotient within a log. The quotient rule of logarithms is useful for expanding and condensing logarithms, along with the product rule and the power rule of logarithms.

Simplifying a logs with a statement with 2 logs and subtraction. Okay? So for this example what I want to do is look at these 2 things individually and then see if there's a way we can throw them together.
So log base 3 of 27, so saying 3 to what power is equal to 27. We know that to be 3. Log base 3 of 9, saying 3 to what power is equal to 9. 3 squared so this is going to be 2. We're subtracting in between so this is just going to be 1. Okay. We can combine these actually and how we do that is by division. so this turns out to be log base 3, 27 over 9. 27 over 9 is 3. Log base 3 of 3, 3 to what power is 3. This is 1. What this gets us is the quotient rule of logarithms and what that tells us is if we are ever dividing within our log, so we have log b of x over y. This is going to be equal to log base b of x minus log base b of y, okay. This is the quotient rule of logarithms. Basically whatever's in the top is going to go first and then we subtract whatever is in the bottom. Okay?
Careful to note that this is not the same thing for log base b of x-y, okay? This only works when we're dividing in the inside of our log. It doesn't work for when we're subtracting. Okay? And we also can go both ways. We'll be able to go from a division in the log to a subtraction or from a subtraction back to division, okay? So that's the quotient rule of logarithms.

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