Introduction to Logarithms An introduction to logarithms
Introduction to Logarithms
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- Welcome to the logarithm presentation.
- Let me write down the word "logarithm" just because
- it is another strange an unusual word, like "hypotenuse",
- and it's good to at least see it once.
- Let me get the pen tool working.
- This is one of my most misspelled words.
- I went to MIT and actually one of the a cappella groups there,
- they were called the Logarhythms.
- Like rhythm, like music.
- But anyway, I'm digressing.
- So what is a logarithm?
- Well, the easiest way to explain what a logarithm is is
- to have first-- I guess it's just to say it's the inverse of
- taking the exponent of something.
- Let me explain.
- If I said that two to the third power-- well, we know that
- from the exponent modules.
- two the third power, well that's equal to eight.
- And once again, this is a two, it's not a z.
- two to the third power is eight, so it actually turns out that
- log-- and log is short for the word logarithm.
- Log base two of eight is equal to three.
- I think when you look at that you're trying to say oh,
- that's trying to make a little bit of sense.
- What this says, if I were to ask you what log base two of
- eight is, this says two to the what power is equal to eight?
- So the answer to a logarithm-- you can say the answer to this
- logarithm expression, or if you evaluate this logarithm
- expression, you should get a number that is really the
- exponent that you would have to raised two to to get eight.
- And once again, that's three.
- Let's do a couple more examples and I think you might get it.
- If I were to say log-- what happened to my pen?
- log base four of sixty-four is equal to x.
- Another way of rewriting this exact equation is to say four to
- the x power is equal to sixty-four.
- Or another way to think about it, four to what
- power is equal to sixty-four?
- Well, we know that four to the third power is sixty-four.
- So we know that in this case, this equals three.
- So log base four of sixty-four is equal to three.
- Let me do a bunch of more examples and I think the more
- examples you see, it'll start to make some sense.
- Logarithms are a simple idea, but I think they can get
- confusing because they're the inverse of exponentiation,
- which is sometimes itself, a confusing concept.
- So what is log base ten of let's say, one million.
- Put some commas here to make sure.
- So this equals question mark.
- Well, all we have to ask ourselves is ten to what power
- is equal to one million.
- And ten to any power is actually equal to one followed by the
- power of-- if you say ten of the fifth power, that's equal
- to one followed by five zero's.
- So if we have one followed by six zero's this is the same thing
- as ten to the sixth power.
- So ten to the sixth power is equal to one million.
- So since ten to the sixth power is equal to one million log base
- ten of one million is equal to six.
- Just remember, this six is an exponent that we raise ten
- to to get the one million.
- I know I'm saying this in a hundred different ways and
- hopefully, one or two of these million different ways that I'm
- explaining it actually will make sense.
- Let's do some more.
- Actually, I'll do even a slightly confusing one.
- log base one / two of one / eight.
- Let's say that that equals x.
- So let's just remind ourselves, that's just
- like saying one / two-- whoops.
- one / two.
- That's supposed to be parentheses.
- To the x power is equal to one / eight.
- Well, we know that one / two to the third power is equal to one / eight.
- So log base one / two of one / eight is equal to three.
- Let me do a bunch of more problems.
- Actually, let me mix it up a little bit.
- Let's say that log base x of twenty-seven is equal to three.
- What's x?
- Well, just like what we did before, this says that x to the
- third power is equal to twenty-seven.
- Or x is equal to the cubed root of twenty-seven.
- And all that means is that there's some number times
- itself three times that equals twenty-seven.
- And I think at this point you know that that
- number would be three.
- x equals three.
- So we could write log base three of twenty-seven is equal to three.
- Let me think of another example.
- I'm only doing relatively small numbers because I don't have
- a calculator with me and I have to do them in my head.
- So what is log-- let me think about this.
- What is log base one hundred of one?
- This is a trick problem.
- So once again, let's just say that this is equal
- to question mark.
- So remember this is log base one hundred hundred of one.
- So this says one hundred to the question mark power
- is equal to one.
- Well, what do we have to raise-- if we have any number
- and we raise it to what power, when do we get one?
- Well, if you remember from the exponent rules, or actually not
- the exponent rules, from the exponent modules, anything to
- the zero-th power is equal to one.
- So we could say one hundred to the zero power equals one.
- So we could say log base one hundred hundred of one is equal to zero
- because one hundred to the zero-th power is equal to one.
- Let me ask another question.
- What if I were to ask you log, let's say base two of zero?
- So what is that equal to?
- Well, what I'm asking you, I'm saying two-- let's
- say that equals x.
- two to some power x is equal to zero.
- So what is x?
- Well, is there anything that I can raise two to
- the power of to get zero?
- So this is undefined.
- Undefined or no solution.
- There's no number that I can raise two to the
- power of and get zero.
- Similarly if I were to ask you log base three of
- let's say, negative one.
- And we're assuming we're dealing with the real numbers,
- which are most of the numbers that I think at this point
- you have dealt with.
- There's nothing I can raise three three to the power of to
- get a negative number, so this is undefined.
- So as long as you have a positive base here, this
- number, in order to be defined, has to be greater than-- well,
- it has to be greater than or equal-- no.
- It has to be greater than zero.
- Not equal to.
- It cannot be zero and it cannot be negative.
- Let's do a couple more problems.
- I think I have another minute and a half.
- You're already prepared to do the level one logarithms module,
- but let's do a couple of more.
- What is log base eight-- I'm going to do a slightly
- tricky one-- of one / sixty-four.
- We know that log base eight of sixty-four would equal two, right?
- Because eight squared is equal to sixty-four.
- But eight to what power equals one / sixty-four?
- Well, we learned from the negative exponent module that
- that is equal to negative two.
- If you remember, eight to the negative two power is the same
- thing as one / eight to the two power.
- eight squared, which is equal to one / sixty-four.
- I'll leave this for you to think about.
- When you take the inverse of whatever you're taking the
- logarithm of, it turns the answer negative.
- And we'll do a lot more logarithm problems and explore
- a lot more of the properties of logarithms in future modules.
- But I think you're ready at this point to do the level one
- logarithm set of exercises.
- See you in the next module.
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