WEBVTT
Kind: captions
Language: en
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Now we're going to learn about how to manipulate logs
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Using the log laws.
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Just the way we have rules of exponents, we have rules of logarithms.
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So..when you have x to the a
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..and you multiple it by x to the b
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..you get x to the a plus b.
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Remember logarithms are powers. So if you have the log of two things multiplied together, that's like when you have x to the a times x to the b.
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That becomes the log of a plus the log of b
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Because you are gonna add the powers, and logarithms are powers.
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What if we have x to the a divided by x to the b?
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Which is x to the a minus b.
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Now were going to have log of a over b.
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Which is log of a minus log of b
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And the third rule, when we have x to the a raised to the b
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We multiply the powers
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So if log of a to the b, that is the same as multiplying b times the log of a
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So those are our three laws. Now let's do a couple of examples..
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If I have the log of 4 times 5
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Which is the log of 20.
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That's going to be the log of 4 times 5, is log of 4 plus the log of 5.
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And if we take out our calculator and you find the log of 20 and then you take the log of 4 and the log of 5 and add them together you will get the log of 20.
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So lets do it.
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We take out the calculator
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..and we have the log of 4
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...is about .6021 and the log of 5 is about .6990. So if we add those together...
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We get 1.3011. By the way these are logs base 10.
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Now whats the log of 20...
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We go to the calculator, and you get 1.3010 but that's for rounding
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So you can see that if you take a look at the log laws..
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The log of two things multiplied together is the first log plus the second one.
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Now another thing we can do is use these log laws to break up something complicated.
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So suppose we have the log of x to the fifth times y to the third
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Well that's the same as saying log of x to the fifth plus log of y to the third
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and that's using our rule, that if two things are multiplied together, it's the log of the first plus the log of the second.
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But we also have our rule that says a log of a number raised to a power is multiplied by the power times the log
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So log of x to the fifth times y cubed is the same as 5 times the log of x plus 3 times the log of y.
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Suppose I had the log of x to the seventh over y to the sixth..
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Well now I can break that into log of x to the seventh minus log of y to the sixth
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because when I am dividing, I subtract the logs.
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And now, if I put the powers infront
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And I get this....
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So here's a nice example..
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If the log of a is 3, and the log of b is 7
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What is the log of a cubed, b squared?
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Well the log of a cubed times b squared..
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Is the log of a cubed plus the log of b squared
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Now I could take the 3 and put it in front..
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and get 3 times the log of a plus 2 times the log of b.
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and now i know, log of a is 3
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and log of b is 7.
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So this gives you 9 plus 14
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..is 23
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So notice..we can take our log laws and break up something complicated and make it into something much simpler.