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Welcome to the logarithm presentation.

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Let me write down the word logarithm just because it is

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another strange and unusual word like hypotenuse and it's

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good to at least, see it once.

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Let me get the pen tool working.

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Logarithm.

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This is one of my most misspelled words.

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I went to MIT and actually one of the a cappella groups there,

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they were called the Logarhythms.

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Like rhythm, like music.

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But anyway, I'm digressing.

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So what is a logarithm?

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Well, the easiest way to explain what a logarithm is is

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to have first-- I guess it's just to say it's the inverse of

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taking the exponent of something.

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Let me explain.

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If I said that 2 to the third power-- well, we know that

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from the exponent modules.

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2 the third power, well that's equal to 8.

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And once again, this is a 2, it's not a z.

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2 to the third power is 8, so it actually turns out that

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log-- and log is short for the word logarithm.

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Log base 2 of eight is equal to 3.

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I think when you look at that you're trying to say oh,

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that's trying to make a little bit of sense.

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What this says, if I were to ask you what log base 2 of

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8 is, this says 2 to the what power is equal to 8?

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So the answer to a logarithm-- you can say the answer to this

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logarithm expression, or if you evaluate this logarithm

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expression, you should get a number that is really the

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exponent that you would have to raised 2 to to get 8.

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And once again, that's 3.

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Let's do a couple more examples and I think you might get it.

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If I were to say log-- what happened to my pen?

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log base 4 of 64 is equal to x.

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Another way of rewriting this exact equation is to say 4 to

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the x power is equal to 64.

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Or another way to think about it, 4 to what

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power is equal to 64?

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Well, we know that 4 to the third power is 64.

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So we know that in this case, this equals 3.

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So log base 4 of 64 is equal to 3.

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Let me do a bunch of more examples and I think the more

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examples you see, it'll start to make some sense.

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Logarithms are a simple idea, but I think they can get

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confusing because they're the inverse of exponentiation,

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which is sometimes itself, a confusing concept.

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So what is log base 10 of let's say, 1,000,000.

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Put some commas here to make sure.

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So this equals question mark.

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Well, all we have to ask ourselves is 10 to what power

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is equal to 1,000,000.

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And 10 to any power is actually equal to 1 followed by the

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power of-- if you say 10 of the fifth power, that's equal

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to 1 followed by five 0's.

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So if we have 1 followed by six 0's this is the same thing

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as 10 to the sixth power.

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So 10 to the sixth power is equal to 1,000,000.

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So since 10 to the sixth power is equal to 1,000,000 log base

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10 of 1,000,000 is equal to 6.

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Just remember, this 6 is an exponent that we raise 10

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to to get the 1,000,000.

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I know I'm saying this in a hundred different ways and

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hopefully, one or two of these million different ways that I'm

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explaining it actually will make sense.

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Let's do some more.

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Actually, I'll do even a slightly confusing one.

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log base 1/2 of 1/8.

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Let's say that that equals x.

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So let's just remind ourselves, that's just

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like saying 1/2-- whoops.

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1/2.

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That's supposed to be parentheses.

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To the x power is equal to 1/8.

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Well, we know that 1/2 to the third power is equal to 1/8.

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So log base 1/2 of 1/8 is equal to 3.

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Let me do a bunch of more problems.

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Actually, let me mix it up a little bit.

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Let's say that log base x of 27 is equal to 3.

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What's x?

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Well, just like what we did before, this says that x to the

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third power is equal to 27.

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Or x is equal to the cubed root of 27.

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And all that means is that there's some number times

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itself three times that equals 27.

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And I think at this point you know that that

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number would be 3.

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x equals 3.

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So we could write log base 3 of 27 is equal to 3.

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Let me think of another example.

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I'm only doing relatively small numbers because I don't have

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a calculator with me and I have to do them in my head.

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So what is log-- let me think about this.

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What is log base 100 of 1?

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This is a trick problem.

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So once again, let's just say that this is equal

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to question mark.

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So remember this is log base 100 hundred of 1.

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So this says 100 to the question mark power

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is equal to 1.

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Well, what do we have to raise-- if we have any number

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and we raise it to what power, when do we get 1?

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Well, if you remember from the exponent rules, or actually not

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the exponent rules, from the exponent modules, anything to

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the 0-th power is equal to 1.

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So we could say 100 to the 0 power equals 1.

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So we could say log base 100 hundred of 1 is equal to 0

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because 100 to the 0-th power is equal to 1.

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Let me ask another question.

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What if I were to ask you log, let's say base 2 of 0?

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So what is that equal to?

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Well, what I'm asking you, I'm saying 2-- let's

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say that equals x.

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2 to some power x is equal to 0.

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So what is x?

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Well, is there anything that I can raise 2 to

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the power of to get 0?

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No.

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So this is undefined.

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Undefined or no solution.

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There's no number that I can raise 2 to the

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power of and get 0.

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Similarly if I were to ask you log base 3 of

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let's say, negative 1.

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And we're assuming we're dealing with the real numbers,

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which are most of the numbers that I think at this point

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you have dealt with.

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There's nothing I can raise three 3 to the power of to

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get a negative number, so this is undefined.

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So as long as you have a positive base here, this

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number, in order to be defined, has to be greater than-- well,

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it has to be greater than or equal-- no.

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It has to be greater than 0.

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Not equal to.

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It cannot be 0 and it cannot be negative.

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Let's do a couple more problems.

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I think I have another minute and a half.

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You're already prepared to do the level 1 logarithms module,

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but let's do a couple of more.

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What is log base 8-- I'm going to do a slightly

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tricky one-- of 1/64.

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Interesting.

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We know that log base 8 of 64 would equal 2, right?

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Because 8 squared is equal to 64.

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But 8 to what power equals 1/64?

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Well, we learned from the negative exponent module that

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that is equal to negative 2.

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If you remember, 8 to the negative 2 power is the same

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thing as 1/8 to the 2 power.

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8 squared, which is equal to 1/64.

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Interesting.

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I'll leave this for you to think about.

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When you take the inverse of whatever you're taking the

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logarithm of, it turns the answer negative.

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And we'll do a lot more logarithm problems and explore

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a lot more of the properties of logarithms in future modules.

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But I think you're ready at this point to do the level 1

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logarithm set of exercises.

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See you in the next module.

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