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Condensing Logarithms - Problem 1

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!

Condensing logarithms. When we condense logarithms what we’re doing is we’re taking a series of logs hat are either added and subtracted from each other and putting them together to make a single more complicated log. So what we do is we use our properties of logarithms to do this. The first one we want to implement is the power rule. We’re used to taking powers down from the term inside the log and bringing it out in front, for condensing we just go the opposite way.

So what we’re going to do is bring all the coefficients up into the exponents. What we have here then is log base 3 of x to the 4th minus log base 3 of y to the 1/2, plus log base 3 of z. Now there’s a number of different ways we can go from here and in general just what I would do is pair a couple first.

Right here, let’s look at the first two, we have the log base 3 of both things so that means we can combine them because they are the same base. We are subtracting in between so that tell us we can use the quotient rule just when it’s subtraction and that would make it division. So combining these two together we would end up with log base 3 of x to the 4th over, and remember you’re fractional exponents, y to the ½ is really the same thing as the square root of y and then plus log base 3 of z of on the outside.

So we combined these two things. We still have to combine the log base 3 of z. Here we are adding, adding is a result of the product rule so that tells us if we put them together into one log we would be multiplying. Ending up with log base 3 x to the 4th, z over root y. So I took this multiplied by z, when you’re multiplying you can put it in the numerator it’s the same exact thing.

By using our properties of logarithms, we took this more complicated logarithm statement and turned it into a single log.

One thing I want to note is if you look at this, the coefficients that are positive so that would be the x term and the z term end up in the numerator and the term that has a negative coefficient, the y term is going to end up in the denominator. You can go through all these steps but hopefully over time as you get used to it you can just realize that if it's positive coefficient it will end up on top, negative coefficient will end up in the bottom.

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