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Common Algebra II Mistakes - Problem 5
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One of the harder areas for students to comprehend is logarithms and there is a couple of really common mistakes that are made when dealing with logs. The first is dealing with expand and condensing and realizing what are equivalent statements.

What I have beside me are 4 different statements dealing with sort of multiplication/addition stuff. There’s only two of them that are actually the same. What you need to know is that if you are multiplying logs there’s really nothing you can do. Likewise if you are adding within logs there’s nothing you can do. The only time you can manipulate things is if you are multiplying inside of a log or adding outside of logs. These two statements are actually equivalent.

Going from if you are to say expand, write it as more than one log, you would go from log xy to log x plus log y. If you were condensing you would go from log x plus log y back to log xy.

Similarly there are common mistakes that occur with division and subtraction. Now I have 5, 6, 5 statements next to me and a couple of them are equivalent. There is a slight change with addition and multiplication to subtraction and division and that is first of all dividing logs, we actually can do that and that's being done by our change of base formula. If we wanted to put log y of x into the calculator, we would want to drop the y down, make it its own log, so these two things are actually equivalent statements.

Just like when we are adding inside of logs we can’t subtract, we can’t do anything if we are subtracting inside of a log. So this statement log x minus y is really just stuck. Then we have the same sort of contrast with expanding and condensing for the division. Log x over y can be expanded to log x minus log y, just like log x minus log y can be condensed it into log x over log y.

In general what I see a lot of is people wanting to expand something like this, log x over y using division or subtraction inside of the logarithm. It’s not the case. If you are dividing inside of your log, it breaks down into subtraction just like if you are multiplying inside of your log it breaks down into addition outside of your log. Just review your log formulas and make sure you’re using the right ones in the right circumstances.

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