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Okay, what I like to do is show you how to evaluate a logarithm.
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And this is a logarithm when we have divisions going on.
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But one thing I notice with this logarithm is, you know, I kind of have this one one,
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and not really too sure what to do.
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So I'm going to show you two different ways we can solve this problem.
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And hopefully one of them is going to work.
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One way I'm going to use division and the other way is that we're going to use multiplication.
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Now a couple of ways, a couple of things we need to remember.
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One thing is, any number as a denominator can always be written as your numerator.
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I can write it back up on the numerator and just put it as to the negative exponent.
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Okay?
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So what I can do for this problem, I'm going to do this two different ways.
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I'm going to first do this by actually changing my five to a negative five to the negative first power.
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So I'm going to say Ln over one times five to the negative first power.
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And here I'm going to solve it as Ln of one over five.
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Now remember Ln we're going to have a base E, right?
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So that's going to be my base.
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So using our two properties, I'm sorry, for logarithms.
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That state, whenever I have a multiplication, I can expand a logarithm by use of addition into two separate logarithms.
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So I can rewrite this as log base, a log, I'm sorry, Ln, not log.
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So I can rewrite this as Ln of one plus at the cheese.
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Ln of five to the negative first power.
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And then here you can also, whenever we're doing division, you can also write a division problem as a set of two multiple logarithms using subtraction.
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So I can write this one as Ln of one minus a normal Ln of five.
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So how are we going to get our answers?
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How are these going to be exactly the same?
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Because it seems like this one is going to be a multiplication, this one is going to be a division.
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How are they going to be the same?
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There's a couple things we need to remember.
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Remember whenever you have a exponent, you can put that exponent in front.
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And then there's another thing.
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Any logarithm, either natural or a regular logarithm, anything,
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whenever you're a bad in logarithm to one, your answer is going to be zero.
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Because think about it this way.
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Log three of nine, that answers two.
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Three raised to what number gives you nine?
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Two. Well, log three of one equals zero.
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Three raised to what number gives you one?
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Well, three to the zero power is going to give you one.
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So any logarithm evaluated for one, no matter what the base is, is always going to give you zero.
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So here I can say Ln of one, E, right?
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Man, this will be either E raised to what number gives you one, zero.
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Plus, I can bring my exponent in front.
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Here, Ln of one is zero minus Ln of five.
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So here all I have is a negative one times Ln of five.
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Well, I know that answer is going to be negative.
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Ln of five. Here, it's a negative. Ln of five.
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So that is just two different ways for you guys to evaluate a logarithm
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when using division.