Honors Algebra II 0809

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*Remember that a logarithm is just another way of writing an exponent

     7^x=15 is equivalent to x=log(7)15

Log/Exponent Equations

1. 2^x=8 x=3

2.(1/2)^x=8  →(1/2)^-3=8  →x= -3

3.4^3x=8^x+1 → (2^2)^3x = (2^3)^x+1 → 2^6x=2 ^3x+3 → 6x =3x+3 [solve for x]→x=1

4.8^5x= 16^3x+4 →(2^3)^5x= (2^4)^3x+4→2^15x=2^12x+16→ 15x=12x+16→3x=16→x=16/3

5.3^x+2=9^x+1 → 3^x+2 =(3^2)^x+1 →3^x+2= 2^2x+2→x+1

Lets talk equation

  We were able to get the same base on each side. However, what about something like 2x=7

Take the LOG of both sides

              Log2^x=  Log7  →xlog2=log7 → x=log7/log2= 2.807

Practice Makes Perfect

10^(2x-3)+4=21→10^(2x-3)=17 →log(10^[2x-3])=log(7)→(2x-3)=log(17)→2x-3=log(17)[add 3 and divide by 2] → x=(log 17 +3)/2

There is no e in log

4-2e^x=-23 → (-2e^x)/2=(-27)/2[change it to positive] → e^x= 13.5 →ln(e^x)=ln(13.5) → xlne=ln(13.5) → x=ln(13.5) [x=2.603]

Our Turn! =]

1. 3^x=14 → x=log(3)14 → log14/log3

2. 10^(3x-1)+5=33 → (3x-1) log10=log 28 → x=(log[28]+1)/3→ x=.816

3. 5e^(x)+10=25 → e=3 → xlne=ln3 →x=ln(3)

4. 4e^(x)+2=18 → 4e^(x)=16 → e^(x)=4 → x=ln(4)

HOMEWORK!!!

pg.505  4-9, 16,17, 19-24, 25-30, (31-41) odds 

We are going to condense logarithms

Condense:

Definition 1- make more dense or compact, reduce the volume or extent of, concentrate

Definition 2- to reduce to a shorter form, abridge

First Examples:

1. log78 – log712 = log7^(3/4)

2. 2logx + logy + log5 = log (5x^2y)

3. ln25 – ln5 – ln1 = ln (25*1/5) = ln(5)

4. 7lnx + 2ln – 3lnz = ln (49x^7/z^3)

More examples! :

1. 2log7x + 1/2log7y = log7x^2 + log7y^1/2 = log7x^2y^1/2

2. 3logx + 2logy – 5/2logz = logx^3 + logy^2 – logz^5/2 = log(x^3y^2/z^5/2)

3. 5(logx – logz) = logx^5 – logz^5 = log(x^5/z^5)

4. 2ln3 – ln67 – ln75 = ln9 -ln67 – ln75 = ln (9/67*75) = Ln (3/1675)

Homework!!!! page 496 9-12,(14-20)evens, (39-45)odds, 46-55, (58-64)evens

first, review the properties of exponents.

x^ax^b = x^a+b
multiplication → addition

xb/x^b = x^a-b
divison → subtraction

same works for logs!

log_buv = log_bu + log_bv

log_b(u/v)=log_bu – log_bv

log _buⁿ = n • log_b u

Expanding Logarithms

Definition 1: to spread out: unfold: develop
Definition 2: to express something more fully or in greater detail

ex:1) log₅ 25x = log₅ 25 +log₅ x
log₅ 25x = 2 + log₅ x

More Examples:

1) log₈ x² = 2log₈X

2) log 6x³ = log 6 + log x³
=log 6 + 3 log x

3) log₆ (17/36)
log₆ 17 – log₆ 36
log₆ 17 – 2

4) ln (10/3)
ln 10 -ln 3

5) log ₅ (625/x ⁷)
log ₅ 625 – log ₅ x ⁷
4 – 7 log ₅ X

6) ln 6x^½ y³
ln 6 + ½ ln x + 3 ln y

7) log (3y ⁴/x³)
log (3y ⁴) – log (x³)
log 3 + 4 log y – 3 log x

8 ) log √x y³ z⁷
log x^1/2 y ^3/2 z ^7/2
1/2logx 3/2logy 7/2 logz

Change of Base Rule

Formula: logc u= log_b u/ log_b c
(b is any base)

change log₅ 7?
log7 = log_b 5= log 7 / log 5
or
=ln 7 / ln 5
5 ^{1.209 = 7}
log₅ 7 = 1.209

More examples:

1) log ₃ 17
log17 / log 3 = 2.579

2) log₂ 125
log125 / log2 = 6.966

3) log₃₂ 7
log 7 / log 32 = .561

HOMEWORK!

page 496
# 5-8, (15-21)odd, 30-37, (38-44)even, (59-65)odd

The first thing we went over in class was how logs can be treated as the “opposite” of exponents to solve for x.

This is formally written as:  g(f(x))=log_bb^x=x  and  f(g(x))=b^log_bx=x

It’s easily shown with a couple examples:

log₅5^x = 5 and  8^log₈x = 8

This can also be applied to problems where the bases aren’t necessarily equal:
       log₃9^x

In this case you can simply rewrite the 9 as (3²) and plug it back into the equation.

      log₃(3²)^x = log₃3^2x; which is simply: 2x

Next we graphed a couple logarithmic functions using a table.

If you remember doing inverse graphs by reflecting over the line y=x then the logarithmic function

f(x)=log₂x is, by definition, the inverse of g(x)=2^x (see following graph):

fooplot only supports natural and common logs (e and 10). If you want to see a graph involving another base use this formula: logx in base n = ln(x)/ln(n)

If you go to page 489, your textbook has a helpful green box explaining how a logarithmic function behaves.

Another method is to make a table of a logarithmic function. The easiest way to do it is to go backwards, starting with a y and determing what the x will be. Let’s do y=log₄x

when y=1, x=4; so the point (4,1) is on the graph

also, y=0, x=1 ; y=1/2, x=2; and there is an asymptope at x=0. Overall it looks like this:

Our homework was p.490: #12; 25-45; 48-55; 65

Look back at exponentials…previously we dealt with equations like 2^x and (1/2)^x

Lets start with y=2^x  and find inverse……. First switch x and y..     x=2^y     Next solve for y but you dont know how to so you have to frown =( .

To deal with exponents in a different manner we use a logarithm   y=b^x        25=5²

                                                         witch is equivalent to          x= log_by    2=log₅25

b= base always    y=b^x  ->  x= log_by

Said ” x equals the log of y base b.” ~Mr. Higgins~ Thought ” b to what power is equal to y.”

First lets do some basic logarithms

Log form           Expoential Form            Value of Log

log₅25                     5^x=125                           3

log₈4096                8^x=4096                          4

log₃₆6                    36^x=6                             1/2

log₇1                       7^x=1                                0

log₅5⁶                            5^x=5⁶                              6 

log100                     10^x=100                          2

log(1/1000)              10^x=10^-3                      -3

Logarithm on a Calculator

1.  log1560=3.193                         10^3.193= 1560

2. log(1/7) = -.845                         10^-.845=(1/7)

3. log.7654 = -.116                        10^-.116= .7654

Now Time For Homework

Page 490 #’s 5-11 16-23 (24-46 evens) 56

P.S. Mr. Higgins told me to say “to remember that logs are a bear and that bears trip over logs in the woods.” lol haha =)

The last two days we have been talking about Factorials!

Factorials can be represented by n!

yeah an (!) mark

Factorials are when you times a number by every whole number smaller than it.

Here are some examples of factorials

5!= 5*4*3*2*1=120

4!=4*3*2*1*=24

3!=3*2*1=6

2!=2*1=2

In facorials 0!=1

Here are some questions involving facorials:

5!4!/3! Since 5! is the biggest factorial you bring it down to 3! which makes the equation 5*4*3!*4!/3!

since there is now a 3! you cross out the  3! the equation looks like 5*4*4!

Since 4!=24 you do 20*24 which equals 480

Here is a problem involing Nth factorial

(n+1)!÷(n+2)!

In this problem since (n+2)! is the biggest you would drop it down to (n+1)!

so now the equation looks like (n+1)!÷(n+2)(n+1)!

Now you can take the (n+1)! out of each equation and you are left with

1/n+2

Factorials are not very hard but you have to make sure you always take the highest factorial in the equation and march it down to the next factorial in the sequence.

Homework=A WORKSHEET!!!!!!!8-)

Previously, we discussed Compound Interest and used terms such as…

Principal        Annually        Monthly        Time        Compounding

The key was that money would compound based on the amount in the account at that time. There were only so many compounds in a given year.

Compound Continuous Interest

*Definition

-Interest which compounds every single moment of every single day over a given time period; however, each new amount is infinitely small.

Calculating the Amount

We use the following formula for compound continuous-

A = Pe^rt   

PERT!!!

P- Principal

r- rate as decimal

t- time in yours

Compound Continuous Example

1. A total of $9,000 is invested at an annual interest rate of 2.5%, compounded continously. Find the balance after 5 years.

a) Annually   A = $9,000e^.025(5)       A = $10,198.34

*To complete this problem in your calculator don’t forget the parentheses-

A = 9000e^(.025*5)

b) Quarterly - to complete use A = P(1+r/n)^nt  

A = $9,000(1+.025/4)^(4)(5)       A = $10,194.37

Another Compound Continous Example

2. A total of $7,000,000 is invested at an annual interest rate of 1.5%, compounded continously. Find the balance after 25 years.

A = 7,000,000e^.015(25)       A = $10,184,939.90

After 250 years.

A = 7,000,000e^.015(250)   A = about 3 million

Example #3

K = thousand

We then watched an amazing clip from Futurama!!!

The boy had a starting balance of $.93 and over 1,000 years with an interest rate of 2.25%, his new balance was 4.3 billion!!!!!

Was it compounded continuous-

.93e^(.0225)(1000) = 5.5  billion    WRONG!

Was it compounded quarterly-

.93(1+.0225/4)^(4)(1000) = 5.2 billion    WRONG AGAIN!

Was it compounded annually-

.93(1+.0225/1)(1)(1000) = 4.3 billion    CORRECT! 

Yea!

Homework = A worksheet!

WARNING WARNING THIS SECTION IS CLASSIFIED AS A “HIGGINS FAV”

Basic facts about compound interest:

*Money deposited in a bank or other investment earn interest at a certain rate.

*Every so often money is coumpounded…which means a new amount is calculated.

*Money can be compounded:

~Monthly-1month-12x a year

~Quarterly-3 months-4x a year

~Semiannually-6 months-2x a year

~Annually-12 months-1x a year

KEYWORDS involving investment:

1.Principal- how much money there is at the start

2.Balance- amount of money ather calculation

EX: You could borrow $100,000 from a bank and with interest you might owe $175,000 back.

EX: If you have a credit card and use $50 its like borowing $50.  But if you don’t pay the bill you at least get 7% interest until the bill is payed.

EX: If you have a credit card and borrow $50,000 they let you pay it back in small amounts like $130/month.  So it takes you 384 months or 32 years to completely pay it off.  The longer it takes you to pay, they raise the rate because you are only paying a small amount at a time.  Sometimes you can get rewards but even for $50,000 you would be lucky to get a $3,000 reward.

EX: The more you pay with credit…the better your credit score gets.  So it makes it easier to get loans whe you need them.

WARNING WARNING HIGGINS EXAMPLE

For college Mr. Higgins borrowed $22,000.  There was 4.7% interest and he owed $28,000 back. 

EX: You can get away without paying off your loans but then you get bad credit and CAN’T get anymore loans.

Money Money Money

The formula for calculating the amount of money is…

A=P(1+r/n)^nt      (^ the norwalk truckers)

A-final amount    

P-principal    

r-rate as a decimal    

n- number of times money is compounded in a year

t-how many years

1) The total of $9,000 is invested at an annual interest rate of 2.5%, compounded anually.  Find the balance after 5 years.

a) Annually A=9000(1+(.025/1))^((1)(5))  A=$10,182.67

b) Quarterly A=9000(1+(.025/4))^((4)(5))  A=$10,194.37

c)$90,000,QuarterlyA=90,000(1+(.025/4))^((4)(5))  A=$101,943.70

d)$90,000,15% interest rate,QuarterlyA=90,000(1+(.15/4))^((4)(5))  A=$187,933.68

2) The total of $12,000 is invested at an annual interest rate of 3%.  Find the balance after 4 years if interest is coumpounded.

a) Quarterly A=12,000(1+(.03/4))^((4)(4))  A=$13,523.91

b) Semiannually A=12,000(1+(.03/2))^((2)(4))  A=$13,517.91

Just remember…more often=more money!

Very easy stuff!  2 more days ahead!

UGH

HW pg 471 #s 57,59-64,67 and pg 483 #s 49-60

At the beginning of class we watched a austin powers video to introduce e.

e is a lot like ∏.

∏ represents a number just like e does.

It is the circumerence/diameter.

It is also irrational.

e= 2.718281828459

It is irrational (never terminates).

It is created using this sequence:

1/0! + 1/1!+ 1/2!+ 1/3!+1/4!…

Uses

Exponential growth and decay and compound interest.

“Bell Curve” in stats and shape in which the cable hangs.

We did some practice problems to show that e is just like using x or ∏.

e^3*e^4=e^7

10e^3/5e^7=2e^-4=2/e^4

(3e^-4x)^2=9e^-8x=9/e^8x

Follow the same rules as u would use for x. You can solve for e by plugging it into the calcualtor. Or if you are really desperate you do it by hand if you want to… (or if you are bored in study hall and dont feel like dividing by 3 anymore.)

The homework was pg 483 4-11,17-32,41-48

An exponential equation is of the form…

y=b^x  where b is called the BASE

Examples:

y=2^x       y=(1/3)^x       y=(27)^x

y=b^x

If b>1 then we have EXPONENTIAL GROWTH

If 0<b<1 then we have EXPONENTIAL DECAY

Graphing an exponential equation       y=2^x

Domain: (-∞, +∞)     Range: (-∞, +∞)

Asymptote- y=0

Y-Intercept- 1

y=3^x-2

Domain: (-∞, +∞)     Range: (-∞, +∞)

Asymptote- y=0

Y-Intercept- 1/9

Homework! Pg. 469 3-6, 9, 19-24, 34