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Condensing Logarithms - Problem 2

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!

Using the properties of logarithms to condense a logarithmic statement. So in this particular statement we have 4 different logs, all added and subtracted together which we want to condense down to a single log. We have number of formulas at our hands, we have our product rule, our quotient rule and our power rule and those are the three main properties that we use when we’re dealing with condensing or expanding logarithms.

We could go through bit by bit and condense two terms and then condense another two and put them all together or I’m going to teach you a little bit of a short cut this time. So what ends up happening is whenever we have a positive coefficient on a term that term is going to end up in the numerator. Whenever we have a negative coefficient that term is going to end in the denominator.

Looking at this I have 2 log base 5 of x. The 2 is positive so therefore I know that the x is going to end up in the numerator. Here I have -3/4 log base 5 of y, I then know that the y is going to end up in the denominator because of this negative sign, and so and so forth with the z and the 3. We also know because of the power rule all these exponents are just going to come up. I’ll do this in two steps for you.

First thing we can do is bring all these exponents coefficients into the exponent spot. What we end up with then is log base 5 x² minus log base 5 y to the ¾ plus log base 5 of z minus log base 5 of 3². We then know that, did I forget my 4, I think I did, didn’t I? So that 4 came up, sorry about that. We now know that anything that has a positive term is going to end up in the numerator; anything with the negative is going to end up in the denominator. So we have a giant log 5.

My x and my z are positive, so those are going to end up in the top, my y and my 3 are negative so they’re going to end up it the denominator, y to the ¾ remember power over root so this is going to be the same thing as the 4th root of y cubed and 3² is the same thing as 9. So basically by knowing that positive coefficients mean numerator and negative coefficients mean denominator, there’s a really easy way to condense this statement down to a single log.

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