Algebra II: Circles and Logarithms

Algebra II

California Standards Test: Algebra II
California Standards Test: Algebra II (Graphing Inequalities
CA Standards: Algebra II (Algebraic Division/Multiplication)
CA Standards: Algebra II
Algebra II: Simplifying Polynomials
Algebra II: Imaginary and Complex Numbers
Algebra II: Complex numbers and conjugates
Algebra II: Quadratics and Shifts
Examples: Graphing and interpreting quadratics
Algebra ||: Conic Sections
Algebra II: Circles and Logarithms
Algebra II: Logarithms Exponential Growth
Algebra II: Logarithms and more
Algebra II: Functions, Combinatorics
Algebra II: binomial Expansion and Combinatorics
Algebra II: Binomial Expansions, Geometric Series Sum
Algebra II: Functions and Probability
Algebra II: Probability and Statistics
Algebra II: Mean and Standard Deviation

Algebra II: Circles and Logarithms 43-48, circles, logarithms, exponents

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Algebra II: Circles and Logarithms

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We're on problem 43.
Which statement describes the graph of
the following equation?
They write x squared plus y squared plus 4x minus 6y minus
3 is equal to 0.
Essentially you want to get this into standard conic
square format-- I don't know what it's officially called--
and you do that by completing the square on both the x and
the y I guess you could call it sub-polynomial.
Then you should be able recognize what kind of conic
section it is.
So let's separate out the x and the y terms. So the x
terms-- I'll do that in blue-- you get x squared plus 4x.
Then we're going to add something that makes it a
perfect square.
You can already guess what we're going to add.
Let's say we're going to add 4, because x squared plus 4x
plus 4 is x plus 2 squared.
So I'm going to write that in magenta just so we remember
that we added it.
This came out of nowhere right now.
So if I added plus 4 here, I'm going to have to add
plus 4 on this side.
This is the equals sign.
All right, then let me do the y terms. I'll do it in this
brown color.
Plus y squared minus 6y.
Then what would add here?
Half of minus 6 is mine 3.
Then if I were to square that, minus 3 squared-- I'll do it
in this magenta color again-- is plus 9.
So I want to add plus 9 to the right-hand side as well.
Of course this is y minus 3 squared, this whole thing
right here.
Then I have that minus 3 right over here.
All right now let's rewrite this term right here.
I added this plus 4 here for a reason so that I could make
this into x plus 2 squared, then plus-- and I added this
plus 9 here for a reason, so I could make
this y minus 3 squared.
Let me just keep the minus 3 there for now.
Minus-- let me do it in the other color-- minus 3 is equal
to 4 plus 9.
4 plus 9 is equal to 13.
Let's add the 3 to both sides of that equation.
So you get x plus 2 squared plus y minus 3 squared is
equal to 16.
Now if you think about it, this is the form of a circle.
x minus a squared plus y minus b squared is equal to r
squared, where the center of the circle is at where?
Whatever the x-value is what makes this equal to 0.
So it's x, it's minus 2.
Then the y-value of its center is what makes this equal to 0.
Minus 2 comma 3, that's its center.
It's radius is the square root of 16.
So the radius is equal to 4.
It can't be negative 4.
You can't have a negative radius.
So the answer is a circle with radius at minus 2 comma
3 and radius 4.
That's choice D.
Problem 44.
What is the solution to the equation 5 to the
x is equal to 17?
OK so that means that we if we take the log base, so five to
the x power is equal to 17.
We could rewrite this in log format.
That means that log base 5 of seventeen is equal to x.
You could read this exactly the same way you read that.
5 to the x is equal to 17.
Of course that's not one of our choices, and this is
fairly useless on your calculator because you don't
have a log base 5 button.
Let me write it this way, x is equal to-- I'm just switching
it-- log base 5 of 17, just because it's easier
for me to deal with.
I prove it in previous Khan Academy logarithm videos, but
there's just a general property.
I'll write the property here.
If x is equal to log base, I don't know, base b of a, that
is equal to log base 10 of a, lowercase a, divided by log
base 10 of b.
You just have to pick the same base.
You could have the base, you could have the natural log of
a divided by the natural log of b.
You could do whatever log you want, but log base 10 is on
your calculator.
So that's why this is convenient.
So we could say this is equal to the log base 10 of 17
divided by the log base 10 of 5.
That should be a choice.
Yep, that's choice D.
Next question.
This problem 45.
I'll write it down, 45.
They say if log base 10 of x is equal to minus 2, what is
the value of x?
So that essentially -- 10 to the minus 2
power is equal to x.
All a logarithm is, it tells you what exponent do I have to
raise the base to to get whatever I'm inputting into
the logarithm function.
Right So if I raise 10 to the minus 2, I get x.
10 to the minus 2, that's 1 over 10 squared is equal to x,
which is 1 over 100 is equal to x.
That's choice C.
I think I could do problem 46 right here.
46.
Which equation is equivalent to log base 3 of
1/9 is equal to x?
OK, read it the exact same way we read question 45.
This is just saying that 3 to the x power is equal to 1/9.
Let's write that.
3 to the x power is equal to 1/9.
Is that one of the choices?
Yeah.
That's choice C.
That was suspiciously easy, I think.
You just have to know the definition of
what a logarithm is.
OK problem 47.
I'll clear the screen.
Let me switch colors.
47.
Oh I'll cut and paste this one because this is interesting.
Let me pull it up a little bit so you can definitely read it.
Because they want us to identify an incorrect step.
OK let me go here.
OK they ask us, which is the first incorrect step in
simplifying log base 4 of 4 over 64.
All right step 1, they said log base 4 4/64 is equal to
log base 4 4 minus log base 4 64.
That's right.
When you divide within the logarithm function or when
you're inputting into the logarithm function, that's
equivalent of taking the log of the top one and subtracting
from that the log of the bottom.
Actually when you divide two numbers of the same base, you
subtract their exponents.
We've proved that in previous videos.
But that's just good enough that you need to know that
that's the correct step right now.
So what's the log base 4 of 4?
4 to what power is equal to 4?
Yeah sure, that's equal to 1.
4 to what power is equal to 64?
Let's see, 4 squared is 16.
4 to the 1 is equal to 4.
4 squared is equal to 16.
4 to the third is 4 times 16, which is 64.
So 4 to the third power is 64.
So this is 3.
So this should be-- let's see, log of 4 minus 16-- right,
this should be a 3 here.
This should be a minus 3.
So this step 2 was the first incorrect step.
They mis-evaluated log base 4 of 64.
They through it was 16 for some reason.
I guess they divided 64 by 4.
But it's actually 4 to the third power of 16, sorry, 4 to
the third power of 64.
So the answer there is 3.
So step 2 is the first incorrect step.
Next question.
All right let me see.
All right, clear image, invert colors.
Let me copy and paste this one too because this is
interesting.
I'll try to copy and paste as many of the
word problems as possible.
This is problem 48.
Let me write that down in the corner.
48, right in that corner.
Jeremy, Michael, Shanan, and Brenda each worked the same
math problem of the chalkboard.
Each student's work is shown below.
Their teacher said that while two of them had the correct
answer, only one of them had arrived at the correct
conclusion using correct steps.
OK they want to which is the completely correct solution.
OK let me just copy and paste those.
So the completely correct solution, OK so let's just
think about what everyone did here.
It looks like this is it dealing with exponents.
All right he had x to third times x to the minus 7.
OK this is already wrong.
Because x to the minus 7 is 1 over x to the 7.
This is one over x to the 7.
So this should be a plus right here.
So he's wrong.
OK Shanan, x to the third times x to the minus 7.
Yep that equals x to the third over x to the positive 7.
Then let's see, 1 over x to the 4.
That is right, right?
Because this is equal to x to the 3 minus 7.
So that's x to the minus 4.
x to the minus 4 is 1 over x to the 4.
So as far as I can tell, Shanan's work
looks pretty good.
Although this was a little weird how she went straight to
1 over x to the 4.
She may just skipped a step.
Michael's work, x to the over x to the minus 7.
OK, he made the same mistake as Jeremy.
X to the minus 7 when you put it in the denominator becomes
x to the positive 7.
Not right.
Then Brenda's work got this step right.
But then this should be x to the minus 4, x
to the minus 4 power.
So this is not correct.
So the only person who got it right was Shanan.
What choices is that?
Shanan is choice C.
Out of time.
See you in the next video.

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