WEBVTT
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Hi, let's take a look at properties of logarithms.
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As I go through them write them down and go over them until you know them well.
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These properties are very important when analyzing log expressions.
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Let's take a look at the first property.
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Log base a of one equals zero.
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Because as you remember the definition of a logarithm is
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the log base a of x equals y means that a to the y equals x.
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We do need to make restrictions on the variables.
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In this case, the base a cannot equal one and must be positive.
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Let's look at an example.
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Log base 4 of 1 equals 0.
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This is because the base 4 when raised to the 0 power will result in 1.
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So you can always rewrite a logarithm as an exponential.
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The base is 4, the exponent is 0, and the result is 1.
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Let's look at the next property.
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Property two is log base a of a equals one.
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And again we have the same restrictions on our variables.
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A must be greater than zero and cannot equal one.
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And if you rewrite this logarithm as an exponential,
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you get a to the first power equals a.
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And so you can see why it's tricky.
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Take a number a, raise it to the first power you'll always get out a.
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Let's look at an example.
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Natural log of e equals what?
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We have our natural log is log base e.
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So this is saying log base e of e equals, of course it will be 1.
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Since e to the first power equals e.
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Okay now let's look at another property.
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The next two properties are inverse properties.
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The first one is a raised to log base a of M equals M.
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And here we have to note again that a cannot equal one,
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a and M must be positive.
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Remember when you are doing a composition,
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a to a power x and log base a of x are inverses of each other
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meaning that they undo each other.
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Whatever value goes in comes out.
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In this case, M.
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So for example,
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if you had say 5 to log base 5 of a number like 14,
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then this will give you 14
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because 14 goes into the logarithm
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and then you plug 5 into the x, they're undoing each other.
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So 14 goes in and then 14 comes out.
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The next property is also an inverse property.
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It says that the log base a of a to the r power equals r
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and again you can see the inverses.
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Log base a of x and a to the x are inverses
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so if r goes in, that's the input then that's the result.
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They undo each other.
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An example of this would be
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the natural log of e to, say 3.2 would equal what?
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Well 3.2 went in so that means, since these are inverses of each other,
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then the result would be 3.2.
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These are very nice to remember because
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they are very easy to once you understand the inverse properties.
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Let's take a look at the next couple of properties.
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Let's look at the fifth property.
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It says log base a of the product MN is
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equal to log base a of M plus log base a of N.
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I think one of the things that might help you here
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if you remember the words that the log of a product equals the sum of the logs.
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Here the product is M times N and that's the same thing as the sum of the logs.
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And again our restrictions here are that a must be greater zero,
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it can not equal one, and M and N must also be greater than zero.
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Let's look at an example.
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If we have log base 6 of 9 plus log base 6 of 4,
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this is the right hand side.
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We have the sum of logs. It will equal the log of the products.
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So we have log base 6 of the product of 9 and 4, which is 36.
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And now we can figure this exact value out
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because what is log base 6 of 36?
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What power do you have to raise 6 to in order to get 36?
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The answer is 2 right.
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So log base 6 of 36 equals 2.
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Let's take a look at the next property.
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The sixth property is log base a of a quotient M/N is
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equal to log base a of M minus log base a of N.
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And as with the property I did a minute ago with the log of a product,
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if you say the words, I think it will help you remember the property.
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This says that the log of a quotient equals the difference of logarithms.
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And this example in this property, M over N is the quotient
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and you can see that the log of the quotient equals the difference of the logs.
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Once again, a is greater than zero, it cannot equal one,
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and the arguments M and N must also be positive.
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Let's look at an example of this.
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If you have, say log base 7 of 14 minus log base 7 of 2,
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how could you use this property and rewrite this?
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You have the right hand side. You've got the difference of logs
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and what does that equal to?
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The difference of logs equals the log of the quotient of 14 and 2.
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And now we know that 14 divided by 2 is 7
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and we can find the exact value of this logarithm.
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7 to what power gives you 7?
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This was an earlier property that equals 1.
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So I can take a difference of logs, rewrite it as a quotient,
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and it is important that you remember here that the
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difference of logs is the log of the quotient, not the quotient of logs.
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So I think the words here will really help you to remember this property.
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Okay let's look at another property.
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Let's look at property number seven.
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Property seven says log base a of M to a power r
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equals r times log base a of M.
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We still have the same restrictions on the base a that we had before.
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A cannot equal one and in this case, both a and M must be positive.
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R can be any real number.
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An example of this would be log base 5 of x cubed is equal to what?
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Well notice we have the left hand side.
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R in this case is 3.
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So what this property says is that we can bring the
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exponent 3 down in front and write it as a product.
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So the 3 comes down in front and so we have then 3 times log base 5 of x.
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Let's look at the last property.
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Let's look at the last property, property number eight.
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It says log base a of M equals log base b of M divided by log base b of a.
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This is called the change of base formula.
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And as before, we have some restrictions on our variables.
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Both a and b cannot equal one and a, b, and M must all be positive.
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This change of base formula is very useful when
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you're trying to find an approximate value for a logarithm.
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Let's look at an example.
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Log base 3 of 20 equals what?
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Well remember logarithms and exponents.
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So I'm looking for what power do I need to raise 3 to in order to get 20.
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Think about that for a moment and what do you think you're going to get?
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Well let's see.
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3 to the second power gives me 9.
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3 to the third power gives me 27.
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So I realize that this is going to be somewhere between 2 and 3.
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Well we can use this change of base formula and go to a different base.
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Now if you know notice your calculator, look at your calculator for a moment.
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It has two log buttons on it.
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Its got the natural log button, ln, and its got the common log button, log.
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So if you're going to use your calculator to approximate,
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you want it to change either the common log base ten or the natural log base e.
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We'll do both just to show you.
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So using the change of base formula,
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we can write this as a common log base 10 of 20.
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The base here is 10, M is 20, a is 3, and we're changing to base 10.
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Remember with common log, you don't see the base
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and so this formula would say it would be over log base 10 again.
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The base is 10 and our a value is 3.
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I can also write this as a natural log of 20 divided by the natural log of 3.
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Now these are all of the exact values.
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If I now use my calculator and approximate this,
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I'll use the last one, the natural log.
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So find your natural log button, hit it.
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Mine automatically gives me an open parentheses.
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So you need to be careful of your calculator, whatever it does, whatever it has.
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You definitely want to make sure you have natural log of 20 by itself.
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And then make sure you close the parentheses or else it will continue to divide.
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Close the parentheses
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and then hit divided by the natural log button again and then I want 3.
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Close the parentheses and then I hit equals.
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And I notice that I'm going to round off at three decimal places
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but you will just follow whatever the directions say in your problem.
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I ended getting 2, I'm going to round to three decimal places.
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I get approximately 2.727.
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And notice before we started this problem we knew
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it was going to be somewhere between 2 and 3.
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So this answer sounds reasonable.
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So this property is very useful when
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you're trying to find an approximate value of a logarithm.
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Now the next examples we'll see are common student errors
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that students make a lot and
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these are the ones you want to be very careful about.
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A lot of students make errors and
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you need to be careful about this.
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Let me point out a couple of errors that students make most often.
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The first one is that log base 4 of a sum.
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Students tend to want to distribute the log through there and
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write this as log base 4 of x plus log base 4 of 1.
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Well this is not equal.
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There is no rule for the log of a sum.
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Remember properties five and six, there is a rule for a log of a product.
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There is a rule for the log a quotient
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but there is no rule for the log of a sum or a difference.
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So don't be tempted to distribute through a sum like this.
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The next one is with a difference.
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For example, if you had the natural log of 2x minus 1.
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That's the log of a difference.
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There is no rule for the log of a difference.
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That is not equal to the natural log of 2x minus the natural log of 1.
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In fact, if you stop and think about it, what is the sum of logs
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that we've talked about earlier?
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The sum of logs is the log of a product not the log of a sum.
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So you need to be very careful. Remember the words will help you.
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The sum of logs is the log of the product.
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The difference of logs is the log of a quotient.
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So there's no rules for the log of a sum or the log of a difference.
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The next error students make is they would be tempted to bring this 3 down.
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It's very tempting because you remember there is
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a property that says you can bring it down in front.
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But that's only when the exponent applies to the argument here,
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not to the whole logarithm.
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This is not equal to 3 times log base 5 of x.
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So don't bring down the 3 there in front when it applies to the whole logarithm.
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It only comes down in front when it's applied to the argument.
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Another error I see is when students have something like this.
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Again, I see that the 3 is applied to the argument but
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you cannot bring the 3 down in front of the whole thing
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because this 3 only applies to the x.
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It doesn't apply to the 6.
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So you can't bring it down in front of the whole thing.
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So be careful. Make note of these common student errors
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and go over all those properties so you know them well.
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Next thing we're going to do is look at using
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properties to expand and condense log expressions.
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We will also use properties of logarithms to expand and condense log expressions out.
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Let's take a look at this next example.
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We want to write as a sum and/or difference of logarithms.
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Express powers as factors.
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You see log of x cubed times square root of x minus 1
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over x plus 2 to the fourth power.
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The base here is 10 because you don't see it
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and remember this is the common log.
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When we look at this, we have to figure out
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what's the big picture and have to expand it out.
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The big picture here is the log of a quotient
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and I remember that property that says
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the log of a quotient equals a difference of logs.
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So I can write this as log of the product
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of these two minus the log of the denominator.
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So I've got the quotient of everything in the numerator minus the denominator.
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The log of a quotient equals the difference of logs.
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Now within this, I see a product.
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I see x cubed times the square root of x minus 1.
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So I have another property that says
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the log of a product equals the sum of logs.
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So I can expand this out as log of x cubed
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plus the square root of x minus 1
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minus log of x plus 2 to the fourth power.
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And we're not quite finished.
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We have a sum or difference but the last step says express powers as factors.
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So there is a property that says we can bring exponents down
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but before we do that,
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we're going to rewrite the square root of x minus 1.
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Do you remember how to write the square root of x minus 1 with no radical symbol?
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That's right. You write it to the 1/2 power.
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So this would be log of x cubed plus
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the log of x minus 1 to the 1/2 minus the log of x plus 2 to the fourth.
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And then our final step is to express the powers as factors using one of our properties.
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It says we can bring the exponent down in front.
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So our final answer will be 3 log of x plus 1/2 log of x minus 1
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minus 4 times the log of x plus 2.
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So here we used three properties.
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We used the log of a quotient. We used the log of a product
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and we used the property that says you can bring the exponent down in front.
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So we expanded this out.
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The last example that we are going to do is the reverse.
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We're going to write it as a single logarithm.
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In this last example, we will use properties of logarithms
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to condense a logarithm into a single logarithm.
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Let's take a look at this example.
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We have log base 3 of x squared minus 36 minus 4 log base of 3 of x plus 6
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and we want to write this as a single logarithm.
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You might be very tempted.
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You see the difference of logs to write as a quotient right now
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but you can't because of this 4.
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That property, remember the difference of logs.
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First of all, the bases have to be the same and they are.
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The coefficients in front of the logs have to be a 1.
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So we can't write it as a quotient yet.
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So we have to think about what we can do with this 4.
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Well remember that property that says you can bring the exponent down.
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Well you can also use that property to bring that exponent up as a power.
0:18:39.500 --> 0:18:45.600
So we can write this as log base 3 of x squared minus 36
0:18:46.500 --> 0:18:52.100
minus log base 3 of x plus 6 to the fourth power.
0:18:53.200 --> 0:18:54.500
So now we have the difference of logs.
0:18:54.600 --> 0:18:56.700
It is very important the coefficients are 1.
0:18:57.300 --> 0:18:59.700
So I have the difference of logs and I remember that property
0:18:59.800 --> 0:19:02.400
that the difference of logs is the log of a quotient.
0:19:02.800 --> 0:19:09.400
So now we can write this as the log base 3 of x squared minus 36
0:19:09.800 --> 0:19:12.800
over x plus 6 to the fourth power.
0:19:13.700 --> 0:19:16.900
So the log, the difference of logs is the log of the quotient
0:19:18.400 --> 0:19:20.400
and notice it says to simplify.
0:19:20.400 --> 0:19:22.200
Can we simplify this expression?
0:19:23.100 --> 0:19:25.700
Well I notice in the numerator I have x squared minus 36
0:19:26.300 --> 0:19:29.900
and the denominator I have x plus 6 raised to the fourth power.
0:19:30.300 --> 0:19:34.500
Well I remember that this can be factored as the difference of squares
0:19:34.500 --> 0:19:39.800
so we can be factored as x minus 6 x plus 6
0:19:40.700 --> 0:19:44.700
and the denominator is x plus 6 to the fourth power.
0:19:46.500 --> 0:19:48.600
So can we simplify this? Yes we can.
0:19:48.600 --> 0:19:52.800
We have a factor of x plus 6 both in the numerator and the denominator.
0:19:53.300 --> 0:19:59.300
So we can finally reduce this and simplify to log base 3 of x minus 6
0:19:59.900 --> 0:20:03.100
over x plus 6 to the third power.
0:20:04.700 --> 0:20:09.500
So now we've written this expanded out form and condensed it into a single logarithm
0:20:09.500 --> 0:20:12.900
and you will need to do this when you're solving log equations later on.
0:20:14.200 --> 0:20:18.200
You need to go over all of these properties that we've gone through very carefully
0:20:18.200 --> 0:20:19.700
and make sure you know them very well.
0:20:20.400 --> 0:20:24.800
They are very important when analyzing log expressions and solving log equations.
0:20:25.000 --> 0:20:28.600
So practice all of the examples, go through them all, and then
0:20:28.600 --> 0:20:30.500
work similar type problems on your own.