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Evaluating a Logarithmic Expression in terms of Known Quantities - Concept

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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!

In order to do problems on evaluating logarithms in terms of known quantities, we need to understand the rules of logarithms such as the product, the quotient and the power rules. When we say evaluating logarithms in terms of known quantities, we mean that we want to be able to rewrite a complex logarithm so that we can recognize certain simpler logarithmic terms within it.

So we're used to using the laws of logarithms to either expend or condense statements okay? So what I have behind me are the two, two of the main properties of logs that we use. The log of a difference we can break it down in subtraction, law of a product we can break it up into addition and how we're used to using this is you know if we're saying expand log 4x squared over square root of z something like that we're used to splitting them up when we have 4 different components inside of our log or multiple components different components in there.
But we can also use this when we are just dealing with a single number okay? And what I mean by that is if you're asked to simplify say the log of 400 okay? Remembering that this is a log base 10 there's that little invisible 10 here, we want to think about how we can simplify this a little bit more and so you want to think of powers that of 10 that go into 400, 100 and then we have it times 4 so this one is equal to log of 100 times 4 we're multiplying so that means we can use our addition property and split it up okay? So this then becomes log 100 plus log 4 okay? Log 100 we know 10 to what power is equal to 100? That's just going to be 2 this statement here is 2 and what we'll actually end up, we end up figuring out is this is equal to 2+log4 okay? So how this actually comes in to play is often times we know information about what a log is so say I knew what log 4 was we could then rewrite this completely using numbers okay we would drop our log altogether so its really convenient when we're trying to rewrite a combination or a number that we know certain factors of without actually knowing what the log is as a whole.

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