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Condensing 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!

Using laws of logarithms to condense a bunch of logs. For this particular example what we’re going to do is condense these three terms down into a single log. The first two can be condensed using the quotient rule. The first thing we have to do is bring our coefficients up to the numerator so we end up with log base 7 of x² minus log base 7 of y to the third and our plus 3 is still hanging out at the end.

We can fairly easily combine these two. The subtraction always turns into division so we end up with log base 7 of x² over y³ and then we’re still left with this plus three. I still want to get this plus three into this log so I just have one statement. The problem is that we somehow, we don’t have any log to bring this into this statement. Whenever we’re combining logs we need to have the same base, so we somehow need to be able to write three as a log base 7.

What we can have is we need a log 7, log base 7 term associated with that. We know that log base 7 of 7 is equal to 1. Remember if the base and what’s inside the log are the same thing, it’s going to be equal to 1. So what we could say then is this is equal to 1, we multiply 1 by 3 and we get 3. So 3 is actually 3 times log base 7 of 7.

Doing the same thing we did over here with our coefficient w can bring that up around and so now we have log base 7 of x² over y³ plus log base 7 of 7³. This is the same exact thing as 3 so we just this term into a log term. Now we are adding which corresponds to multiplication so we can take this 7³ and bring it into the same log, leaving us with log base 7, 7³, x² over y to the third. You could plug it into your calculator figure out 7³ or you could just leave it in this form as well.

Condensing a bunch of logs and also dealing with a constant term, all you need is to turn the constant term into the same log, just typically by doing log base, base of the same number that you’re already dealing with.

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