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Logarithm properties
- Introduction to logarithm properties
- Introduction to logarithm properties (part 2)
- Logarithm of a Power
- Sum of Logarithms with Same Base
- Using Multiple Logarithm Properties to Simplify
- Operations with logarithms
- Change of Base Formula
- Proof: log a + log b = log ab
- Proof: log_a (B) = (log_x (B))/(log_x (A))
- Proof: A(log B) = log (B^A), log A - log B = log (A/B)
- Logarithmic Equations
- Solving Logarithmic Equations
- Solving Logarithmic Equations
- Logarithmic Scale
- Richter Scale
Introduction to logarithm properties Introduction to the first two logarithm properties.
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- Welcome to this presentation on logarithm properties.
- Now this is going to be a very hands-on presentation.
- If you don't believe that one of these properties are true
- and you want them proved, I've made three or four videos
- that actually prove these properties.
- But what I'm going to do is I'm going to show the properties and then show you how they can be used.
- It's going to be little more hands-on.
- So let's just do a little bit of a review of just what a logarithm is.
- So if I say that "a"... oh that's not the right...
- Let's see.
- I want to change-- There you go.
- Let's say I say that a-- Let me start over.
- a to the b is equal to c.
- So if we-- a to the b power is equal to c.
- So another way to write this exact same relationship instead
- of writing the exponent, is to write it as a logarithm.
- So we can say that the logarithm base a of
- c is equal to b.
- So this are essentially saying the same thing, they just have different kind of results.
- In one, you know a and b and you're kind of getting c.
- That's what exponentiation does for you.
- And the second one, you know a and you know that when
- you raise it to some power you get c.
- And then you figure out what b is.
- So they're the exact same relationship, just stated
- in a different way.
- Now I will introduce you to some interesting
- logarithm properties.
- And they actually just fall out of this relationship and
- the regular exponent rules.
- So the first is that the logarithm-- Let me do
- a more cheerful color.
- The logarithm, let's say, of any base-- So let's just call
- the base-- Let's say b for base.
- Logarithm base b of a plus logarithm base b of c-- and
- this only works if we have the same bases.
- So that's important to remember.
- That equals the logarithm of base b of a times c.
- Now what does this mean and how can we use it?
- Or let's just even try it out with some,
- well I don't know, examples.
- So this is saying that-- I'll switch to another color.
- Let's make mauve my-- Mauve-- I don't know.
- I never know how to say that properly.
- Let's make that my example color.
- So let's say logarithm of base two of-- I don't know --of eight
- plus logarithm base two of-- I don't know let's say --thirty-two.
- So, in theory, this should equal, if we believe this
- property, this should equal logarithm base two of what?
- Well we say eight times thirty-two.
- So eight times thirty-two is two hundred and forty plus sixteen, two hundred and fifty-six.
- Well let's see if that's true.
- Just trying out this number and this is really isn't a proof.
- But it'll give you a little bit of an intuition, I think, for
- what's going on around you.
- So log-- So this is-- We just used our property.
- This little property that I presented to you.
- And let's just see if it works out.
- So log base two of eight.
- two to what power is equal to eight?
- Well two to the third power is equal to eight, right?
- So this term right here, that equals three, right?
- Log base two of eight is equal to three.
- two to what power is equal to thirty-two?
- Let's see.
- two to the fourth power is sixteen.
- two to the fifth power is thirty-two.
- So this right here is two to the-- This is five, right?
- And two to the what power is equal to two hundred and fifty-six?
- Well if you're a computer science major, you'll
- know that immediately.
- That a bite can have two hundred and fifty-six values in it.
- So it's two to the eighth power.
- But if you don't know that, you could multiply it out yourself.
- But this is eight.
- And I'm not doing it just because I knew that three
- plus five is equal to eight.
- I'm doing this independently.
- So this is equal to eight.
- But it does turn out that three plus five is equal to eight.
- This may seem like magic to you or it may seem obvious.
- And for those of you who it might seem a little obvious,
- you're probably thinking, well two to the third times two to the
- fifth is equal to two to the three plus five, right?
- This is just an exponent rule.
- What do they call this?
- The additive exponent prop-- I don't know.
- I don't know the names of things.
- And that equals two to eight, two to the eighth.
- And that's exactly what we did here, right?
- On this side, we had two the third times two to fifth essentially,
- and on this side you have them added to each other.
- And what makes the logarithms interesting is and why-- It's
- a little confusing at first.
- And you can watch the proofs if you really want kind of
- a rigorous-- my proofs aren't rigorous.
- But if you want kind of a better explanation
- of how this works.
- But this should hopefully give you an tuition for why this
- property holds, right?
- Because when you multiply two numbers of the
- same base, right?
- Two exponential expressions of the same base, you
- can add their exponents.
- Similarly, when you have the log of two numbers multiplied
- by each other, that's equivalent to the log of each
- of the numbers added to each other.
- This is the same property.
- If you don't believe me, watch the proof videos.
- So let's do a-- Let me show you another log property.
- It's pretty much the same one.
- I almost view them the same.
- So this is log base b of a minus log base b of c
- is equal to log base b of-- well I ran out.
- I'm running out of space --a divided by c.
- That says a divided by c.
- And we can, once again, try it out with some numbers.
- I use two a lot just because two is an easy number to
- figure out the powers.
- But let's use a different number.
- Let's say log base three of-- I don't know --log base three of--
- well you know, let's make it interesting --log base three of
- one / nine minus log base three of eighty-one.
- So this property tells us-- This is the same thing as--
- Well I'm ending up with a big number.
- Log base three of one / nine divided by eighty-one.
- So that's the same thing as one / nine times one / eighty-one.
- I used two large numbers for my example, but
- we'll move forward.
- So let's see.
- nine times eight is seven hundred and twenty, right?
- nine times-- Right.
- nine times eight is seven hundred and twenty.
- So this is one / seven hundred and twenty-nine.
- So this is log base three over one / seven hundred and twenty-nine.
- So what-- What does-- three to what power is equal to one / nine?
- Well three squared is equal to nine, right?
- So three-- So we know that if three squared is equal to nine, then we
- know that three to the negative two is equal to one / nine, right?
- The negative just inverts it.
- So this is equal to negative two, right?
- And then minus-- three to what power is equal eighty-one?
- three to the third power is twenty-seven.
- So three to the fourth power.
- So we have minus two minus four is equal to-- Well, we could
- do it a couple of ways.
- Minus two minus four is equal to minus six.
- And now we just have to confirm that three to the minus sixth
- power is equal to one / seven hundred and twenty-nine.
- So that's my question.
- Is three to the minus sixth power, is that equal to seven-- one / seven hundred and twenty-nine?
- Well that's the same thing as saying three to sixth power is
- equal to seven hundred and twenty-nine, because that's all the negative exponent
- does is inverts it.
- Let's see.
- We could multiply that out, but that should be the case.
- Because, well, we could look here.
- But let's see.
- three to the third power-- This would be three to the third power
- times three to the third power is equal to twenty-seven times twenty-seven.
- That looks pretty close.
- You can confirm it with a calculator if you
- don't believe me.
- Anyway, that's all the time I have in this video.
- In the next video, I'll introduce you to the last
- two logarithm properties.
- And, if we have time, maybe I'll do examples with
- the leftover time.
- I'll see you soon.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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