Logarithms / Exponential Functions

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Exponential Functions and Logarithmics Functions
are INVERSES of eachother.
y = log base 10 of x

MEANS:
y is the power on 10
that gives x as an
answer.

RULES which can be used with logs:

Some examples follow:

Puzzles - using for extra credit

 

I would like to share some puzzles with the Math Teacher Community.
Please feel free to use any or all of the problems.

I believe my unique method used to award the points will be of interest. When I gave any extra credit points for a puzzle I would offer the same number of points as there was students in the class. (25 students could get 25 points.) Too many points to offer? Well there was a catch. Everyone who got the question correct would share equally in the total number of points available. (Thus, if only 2 got the question correct, then they each got about 25/2 [12 or 13] points. If 5 students were correct, then each student would get only 5 points each. If 13 students were right, then each one got about 2 points. This wrinkle in the distribution of points slowed down the sharing of correct answers within my class!

Puzzle #1
Sometimes called a Word Arithmetic Problem:
You must find the distinct digit that replaces a letter, each time that it appears,
so as to satisfy the arithmetic problem presented.
Given:
SMAX + SMAX + SMAX + SMAX = XAMS
What will "XMAS" equal?

Puzzle #2
The sum of all the positive integer factors of
a certain prime number is 32.
What is this prime number?

Puzzle #3
If a quart and a half of egg nog costs $6.30,
how much would two pints cost?

Puzzle #4
Find the SUM of the following ROMAN NUMERALS:
VII + XIV
Give your answer in Roman Numerals.

Puzzle #5
A REGION bounded by the X-AXIS and the
LINES Y = M X + 4 , and X = 1 and X = 4
has an AREA of 7 square units.
What must be the value for M?

Puzzle #6
Successive DISCOUNTS of 10% and then 20%
are equivalent to what SINGLE DISCOUNT?

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ANSWERS:
1) 8172
2) 31
3) $4.20
4) XXI
5) -2/3
6) 28 %
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Puzzle #7
A father has a son.
As of now, their ages total 100.
The father is three times as old as
the son was when the father was
ten years older than the son is now.
How old is the father?
.........................................................
Answer: The father is now 63.
(The best method for solving this
problem is "Guess and Check".
The equations that evolve using
an Algebraic approach are very
messy!)
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Here is good Christmas Handout:
Separate the SNOWMEN!
(Use only 6 straight lines.)

LOOK BELOW FOR THE SOLUTION

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Math in Nature - Fibonacci

Fibonacci Sequence:

0,1,1,2,3,5,8,13,21,34,...

(add two adjacent terms to get the following term!)

There is a great website to see

Fibonacci Numbers in Nature:

http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

1) Is 13 really that unlucky?

2) Why do we consider 21 as ADULTHOOD?

3) Why do we use 3 by 5 cards for notes?

4) Is a 5 by 8 inch photo visually appealing to you?

5) The Mountain Laurel Blossom (found in the Smokies)

has the shape of a Pentagon! (5 sided)

Maybe SIGNS are not considered Nature,
but we find them in our world.
First let's review the names of polygons:

NOW A FEW SIGN EXAMPLES:

Below is the bottom of a stalk of CELERY.
(Check out the SYMMETRY!)

Fibonacci Numbers and Nature :

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

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MATRICES

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Arranging numbers in a Rectangular Form
is an integral part of a spreadsheet.
Mathematics has been using this style
before computers existed. The MATRIX
is a RECTANGULAR ARRAY OF NUMBERS.

The DIMENSION of a Matrix has two
numbers the first reflects the
number of rows and the second number
represents the number of columns.
Here is a matrix with
three rows and two columns:


Here is a link to a great Matrix Site:


INVERSE MATRICES
(only exist for a SQUARE MATRIX)
are used to solve Simultaneous Equations

A=[1 2 3; 4 1 2; 5 4 2]

A =
1 2 3
4 1 2
5 4 2

inv(A)=
-6/31 8/31 1/31
2/31 -13/31 10/31
11/31 6/31 -7/31

Solving One Variable Equations by UNWRAPPING

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Suppose we are trying to
SOLVE: 2x + 3 = 13 Dilemma: What should be done first?
Should we divide by 2 or subtract 3?.
(One explanation is to follow "PMDAS in REVERSE!")
Subtract 3 from both sides,
then divide both sides by 2.

Here is another explanation:
Let's pretend that x is like a box.
Suppose we put 5 into the x-box.
Let's think of the expression
"2x + 3"
as a wrapped package.
We need to unwrap the package
to look inside the x-box!

"X" is wrapped first with
paper having
"MULTIPLY by 2"
as the pattern on the paper.

Then on top of the paper we have
a RIBBON with
"ADD 3"
as the pattern on the ribbon.

To unwrap "2x+3" we first
"CUT the RIBBON" ("SUBTRACT 3")

and our second task is to
"TAKE-OFF the paper" ("DIVIDE by 2").

Performing these same tasks on the
right side of the equation will produce:

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GENERAL HINTS for Praxis Test

1) No penalty for guessing. GUESS!!!
2) Use the graphing calculator as often as possible.
3) Sometimes you can just plug-in each option to see
which one is correct.
4) More difficult problems could be solved by
looking at a more simplified form.

5) If you have no idea, then choose the one

that is most likely!
OPTIONS: 1/2, -1/2, 5, -5, 4

Since 3 options are positive and 3 are whole numbers,

then pick the POSITIVE 5 since there are two 5's!

6) If the question asks for the largest, then pick the

one OPTION that has the largest ABSOLUTE VALUE!

OPTIONS: -5, -4, -2, 3, and 4 CHOOSE -5

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INVERSE FUNCTIONS

All functions have the property where each x has one y-partner.
(Graphically: It passes the "VERTICAL LINE TEST".)
If a certain function also has each y with only one x-partner
(Thus, passing a "HORIZONTAL LINE TEST"), it has an INVERSE!

Prefixes like INV or ARC are sometimes used to denote inverse.
Another form looks like f to the -1 power(Not to be confused with RECIPROCAL).



You might say that the INVERSE
undoes what the orginal function did!

To find the inverse of a function f(x)
switch x and y and THEN solve for y.


f(x) = x cubed
switch
x = y cubed
solving for y
y = cube root of x
y = x raised to 1/3 power

(We had to invent the symbols for cube root and 1/3 power.)
The graph of the inverse of f(x) we can graph f(x) and then
rotate this graph through y = x (thus, switching x and y)

Inverse Trig:

INVERSE TRIG:

How to find the partners

for 22 degrees.

TRIGONOMETRY - Temperature in Cincinnati

Take the xy-plane and
draw a circle of radius ONE.
The equation for this circle is
x^2 + y^2 = 1.
If you wish to use the graphing
calculator to draw the circle you
will need to split it into two
functions. (It is not a function itself!)
y1 = ( 1 - x^2)^(1/2)
y2 = (-1)( 1 - x^2)^(1/2)
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Draw an angle "SMILE" in a
STANDARD POSITION
(start with a side along the x-axis and then rotate
counter-clockwise a certain number of degrees).
The side along the x-axis is called the initial side
and the other side is the terminal side.
The TERMINAL SIDE will intersect the unit circle
at an ordered pair.
This ordered pair is by definition:
the COSINE and SINE of the ANGLE drawn.
Thus, a 30 degree angle will intercept
the circle at the ordered pair (.5, .8660254).