Algebra II
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California Standards Test: Algebra II
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California Standards Test: Algebra II (Graphing Inequalities
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CA Standards: Algebra II (Algebraic Division/Multiplication)
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CA Standards: Algebra II
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Algebra II: Simplifying Polynomials
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Algebra II: Imaginary and Complex Numbers
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Algebra II: Complex numbers and conjugates
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Algebra II: Quadratics and Shifts
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Examples: Graphing and interpreting quadratics
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Algebra ||: Conic Sections
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Algebra II: Circles and Logarithms
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Algebra II: Logarithms Exponential Growth
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Algebra II: Logarithms and more
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Algebra II: Functions, Combinatorics
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Algebra II: binomial Expansion and Combinatorics
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Algebra II: Binomial Expansions, Geometric Series Sum
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Algebra II: Functions and Probability
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Algebra II: Probability and Statistics
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Algebra II: Mean and Standard Deviation
Algebra II: Logarithms and more 53-59, Logarithms and functions
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- We're on problem 53.
- Which table below correctly describes the points of the
- exponential function f of x is equal to 3 to the
- minus x minus 2?
- They all input the same x-values as far as I can tell.
- Yeah, they all input the same x-values.
- So let's just figure out which choice works.
- So if we take f of minus 2, that's the first
- one they give us.
- f of minus 2, that is equal to 3 to the minus x.
- Well, x is now minus 2, so minus minus 2 minus 2.
- So what's minus minus 2?
- Well, that's the same thing as 3 squared minus 2, so that is
- equal to 9 minus 2 is equal to 7.
- Did I get that right?
- Do you see any of these choices?
- Well, only one of them has that. f of minus
- 2 is equal to 7.
- That's not right.
- That's not right.
- That's not right.
- Well, here when x is minus 2, f of x is equal
- to 7, choice D.
- We could have just evaluated the other ones and we probably
- would have agreed with all of these numbers as well.
- Well, that was a lot of copying and pasting for a
- fairly straightforward problem.
- Next problem.
- I already had it there waiting for us.
- log base 6 of 40.
- So, in general, and just inspecting it, I think they
- just want you to figure out how do you figure out a
- logarithm of any arbitrary base of some number if you
- only have a log base 10 function, which tends to be
- the case in your calculator.
- I proved this in a Khan Academy video, but the general
- idea is if I have log base a of b, that this is equal to
- log base 10 of b divided by log base 10 of a.
- Actually, these both don't have to be 10.
- They just have to be the same base.
- 10 tends to be convenient.
- The other base that tends to be convenient is e.
- But anyway, let's use that property
- to solve this problem.
- So log base 6 of 40 is equal to log base 10 of 40 divided
- by log base 10 of 6, which is choice D.
- All right, next problem.
- Copy and paste it.
- Just so you know, I don't like talking while I'm copying and
- pasting, because I think it slows my computer down and
- then I start sounding like Optimus Prime.
- All right, Jonathan wrote the equation log base 6 of x minus
- 4 is equal to 0 on the board.
- He needs one clue for problem solving.
- Which fact provides the correct information that he
- needs to solve the equation?
- Well, I don't know if he needs any information.
- This statement is essentially saying that 6 to the 0th power
- is equal to x minus 4.
- Then you could add 4 to both sides, and you would get 6 to
- the 0th power plus 4 is equal to x.
- I guess if he didn't know the definition already of 6 to the
- 0th power, he would have to know that to know that it
- equals 1, so 1 plus 4 is equal to x, which is 5.
- So I guess he has to know that 6 to the 0th power is a piece
- of information.
- Well this one wouldn't have been useful.
- 6 to the first power, that's not even useful even if you
- didn't know it.
- 4 minus 4 isn't useful even if you didn't know if.
- But 6 minus 4, even that's not useful.
- Yes, choice A, he had to know 6 to the 0th
- power is equal to 1.
- All right.
- Let's see, they have a couple more.
- I'm going to copy and paste all of them at the same time.
- I'm getting bold.
- All right, like that.
- Let me clear all of this: clear image, image, invert
- colors, and I'm back in business.
- There you go.
- All right, I never know whether I should use a
- dark-colored or light-colored marker, because over here, the
- light-colored ones look good, but on the white, the darks.
- So who knows?
- Magenta tends to be good, maybe purple.
- What is the value of log base 3 of 27?
- So that's just asking 3 to what power is equal to 27?
- 3 to the first is 3.
- 3 squared is 9.
- 3 to the third power is equal to 27.
- So log base 3 of 27 is equal to 3.
- They, I guess, just want you to know what the definition of
- a logarithm is in that question.
- I'm noticing that there isn't necessarily a progression of
- difficulty in the problem.
- They kind of flip between an easy one and a hard one.
- Who knows?
- They're just a grab bag of questions.
- If log base 2-- no, not log base 2.
- They're saying log of 2 is approximately equal to 0.301.
- When people just write a log without a base, you usually
- assume it's base 10.
- It's kind of like what you'd get in your calculator.
- If you just press log on your calculator, it's log base 10.
- So 10 to the 0.3 power is equal to 2.
- If log base 10 of 2 is approximately equal to 0.301,
- and log base 10 of 3 is approximately equal to 0.477,
- what is the approximate value of log of 72?
- So let's see, I think we just have to do some prime
- factorization here.
- So 72, that's equal to the log-- let's think about it.
- That's the log of-- let me think about it.
- This is 36 times 2.
- Well, I don't know if that even helps us.
- 3 to the third, that's 27.
- This is 9 times 8.
- So this is log of 3 to the third times 2
- squared; is that right?
- So this is log of 9.
- I made a mistake.
- Ignore that.
- So this is the same thing as the logarithm of 9 times 8.
- Fair enough.
- So that's the same thing as the log of 9
- plus the log of 8.
- Fair enough.
- Well, that's the same thing as the log of 3 squared plus the
- log of 2 to the third.
- 2 to the third is equal to 8.
- Let's keep going.
- I think you see where it's going.
- 3 squared is 9.
- OK, so log of 3 squared, that's the same thing as 2
- times the log of 3.
- The log of 2 to the third power, that's the same thing
- as 3 times the log of 2.
- We know what the log of 3 is and we know the log of 2 is.
- The log of 3 is 0.477, so we have 2 times 0.477 plus 3
- times the log of 2, plus 3 times 0.301.
- So 2 times 0.477-- let me do it here.
- So 0.477 times 2.
- 2 times 7 is 14 plus 1 is 15.
- 2 times 4 is 8 plus 1 is 9, three numbers behind the
- decimal point, 0.954.
- Then 3 times point 0.301, that's easier.
- That's 0.903.
- If you add them together, I can already tell you, it's
- going to be that right there.
- It's going to be 1.857.
- It's choice D.
- Problem 58.
- If x is a real number, for what values of x is the
- equation 3x minus 9 over 3 is equal to x minus 3 true?
- All right.
- Well, if x is a real number, let's just
- play around with it.
- Maybe it's just a truism.
- Well, I know it is a truism.
- Look at it.
- Let me write it.
- 3x minus 9 over 3 is equal to x minus 3.
- Let's multiply both sides of this equation by 3.
- You'll get 3x minus 9 is equal to 3x minus 9.
- So we can stop there.
- So if I take 3 times a number and subtract 9 from it and if
- I said that's the same thing as taking 3 times a number and
- subtracting 9 from it, well, then x could be any number.
- So it works for all values of x.
- You can keep solving.
- Let's just add 9 to both sides.
- You get 3x equal to 3x.
- Divide both sides by 3, you get x is equal to x.
- Well, that's true for any real number x.
- It's equal to itself.
- So that works for all numbers x.
- Let's see what else they have to throw at us.
- All right, let's do one more.
- I've copied, and now I have pasted.
- On a recent test, Jeremy wrote the equation x squared minus
- 16 divided by x minus 4 is equal to x plus 4.
- Which of the following statements is correct about
- the equation he wrote?
- So let's play with it a little bit. x squared minus 16 over x
- minus 4 is equal to x plus 4.
- So immediately when you see something like this, in order
- to do this, what he had to say is that x squared minus 16 is
- x plus 4 times x minus 4.
- That's over x minus 4.
- Then to simplify from this to that, he essentially canceled
- out both of these things.
- That's fine.
- That's fine, assuming that this number
- down here is non-zero.
- 0 over 0 is undefined.
- Anything over a 0 is undefined.
- His simplification is true as long as x does not equal 4.
- Because if x equaled 4, then you would have an 8 times a 0
- divided by a 0, and 8 times a 0 divided by a 0 is not
- defined, so this doesn't work when x is equal to 4.
- If you think about it, if I were to graph this equation or
- his original equation, it would look just like the
- equation of x plus 4, except it would have a hole at 4
- because it's not defined at x is equal to 4.
- So they're not exactly the same.
- The graphs would be exactly the same except at the point x
- is equal 4.
- So the point is b.
- All right.
- I'm out of time.
- See you in the next video.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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