1. NATURES OF LOGARITHM
a to the power of m times a to the power of n equals a to the power of m plus n in bracket.
a to the power of m over a to the power of n equals a to the power of m minus n in bracket.
log base a of b equals n equivalent b equals a to the power of n
example :
question : log base g of a times b in bracket equals…
answer :
log base g of a equals x equivalent a equals g to the power of x
log base g of b equals y equivalent b equals g to the power of y
a times b equals g to the power of x in bracket times g to the power of y in bracket
a times b equals g to the power of x plus y in bracket
log base g of a times b in bracket equals log base g of g to the power of x plus y in bracket equals x plus y in bracket times log base g of g. Note : log base g of g equals 1. So, log base g of a times b in bracket equals x plus y. So, log base g of a times b in bracket equals log base g of a plus log base g of b.
a over b in bracket equals g to the power of x in bracket over g to the power of y in bracket, equivalent a over b in bracket equals g to the power of x minus y in bracket, equivalent log base g of a over b in bracket equals x minus y in bracket, equivalent log base g of a over b in bracket equals log base g of a minus log base g of b.
log base g of b to the power of n in bracket equals log base g of a times a times a times a times a no limit in bracket equivalent log base g of a to the power of n in bracket equals n times log base g of a.
2. HOW TO FIND THE VALUE FROM PHI NUMBER
Calculating approach phi can be elaborated as follows. Since is ahead known that by the ratio of area of circle to its distance square and ratio circle the circle with its diameter is constant. But, initially not yet been known that second of the constanta is of equal. Ancient book use the different constanta to be second of the ratio. Calculation Phi draw attention since pre-christian epoch ( about 1650 SM, in ancient Egypt). Since then hitherto a lot of all saintis conducting calculation of either through analytic and also by using computer. In modern epoch now calculation phi have time to be made by one of test to measure sophisticated and also a logarithm.
Searching number phi got from measurement of area of circle that is open bracket 8 over 9 times d close bracket to the power of 2; by d is diameter, and d equals 2 times r; by r is radius. So that got by equation of area of circle (A) equals open bracket 8 over 9 times 2 times r close bracket to the power of 2, equivalent A equals 64 over 81 times 4 times r to the power of 2, equivalent A equals 256 over 81 times r to the power of 2, equivalent 3,16 times r to the power of 2. Considering formula look for the area of circle equals phi times r to the power of 2, so, value of phi is 3,16.
3. HOW TO DETERMINE THE ABC FORMULA
To determine the ABC formula, we have had the common square equation, that is a times x to the power of 2 plus b times x plus c equals 0. Afterwards the equation is we alter in the form of perfect square equation, coefficient of x to the power of 2 turned into 1, so, the equation become x to the power of 2 plus open bracket b over a close bracket times x plus open bracket c over a close bracket equals 0. Than, left internode and joint right added with open square bracket b over open bracket 2 times a close bracket close square bracket to the power of 2, so, the equation become x to the power of 2 plus open bracket b over a close bracket times x plus open square bracket b over open bracket 2 times a close bracket close square bracket to the power of 2 plus open bracket c over a close bracket equals open square bracket b over open bracket 2 times a close bracket close square bracket to the power of 2. Than, alter x to the power of 2 plus open bracket b over a close bracket times x plus open square bracket b over open bracket 2 times a close bracket close square bracket to the power of 2 become open square bracket x plus open bracket b over 2 times a in bracket close bracket close square bracket to the power of 2, and move the internode c over a, so, the equation become open square bracket x plus open bracket b over 2 times a in bracket close bracket close square bracket to the power of 2 equals b to the power of 2 in bracket over 4 times open bracket a to the power of 2 close bracket plus c over a. than alter b to the power of 2 in bracket over 4 times open bracket a to the power of 2 close bracket plus c over a become open bracket b to the power of 2 minus 4 times a times c close bracket over open bracket 4 times open bracket a to the power of 2 close bracket. Than alter the mentioned equation become x plus open bracket b over 2 times a in bracket close bracket equals plus minus square root of open bracket b to the power of 2 minus 4 times a times c close bracket over 4 times a to the power of 2 close bracket. The last, move b over open bracket 2 times a close bracket to joint right, the equation become x equals negative b over 2 times a plus minus 1 over 2 times a times square root of open bracket b to the power of 2 minus 4 times a times c close bracket, so, x equals negative b plus minus square root of open bracket b to the power of 2 minus 4 times a times c close bracket over 2 times a.
4. HOW TO PROOF A SQUARE ROOT OF 2 IS IRRATIONAL NUMBER
To prove the square root of 2 is irrational number, provable through contradiction verification. Verification of through contradiction at ancient Greek era is verification that square root of 2 representing irrational number ( cannot be expressed by as integer comparison). This statement is provable by assuming on the contrary that square root of 2 is rational number, so that can be expressed by as comparison of integer a divided by b in bracket in simplest fraction. But if a divided by b in bracket equals square root of 2, hence a times 2 equals 2 times b times 2. This means a times 2 is even number. Because square from anomalous number not possible to be even, hence a is even number. Because a divided by b in bracket is simplest fraction of anomalous sure b ( even divided even fraction cause still can be made moderate). But because a is even number ( assume the its 2 times r meaning is a times 2 open bracket 4 times r times 2) is fold number 4, and b times 2 is fold number 2 ( even). Matter of this means b also represent the even number, and this represent the contradiction to previous conclusion that anomalous sure b. Because assumption of early that square root of two is rational result the happening of contradiction, the assumption wrong surely, and denied ( that square root of 2 is irrational) representing real correct statement.
Senin, 06 April 2009
EXPRESSING THE VIDEO WHICH I HAVE SEEN IN ENGLISH PART 2 LESSON BY RIZAL AHMAD 08301244039 PMNR08
VIDEO 1
DO YOU BELIEVE
After seeing video entitling do you believe, I embrace some of words said by the remarkable speaker is which still occupy the class of fifth elementary school
Do you believe?
I can do anything, dream anything, and become anything.
I can do anything, dream anything, and become anything because you believe me.
Let’s me ask you a question that’s easy.
Do you believe on my classmate.
Do you believe that every singles can graduate where classmate, school, and collage.
You better, because next week, I join with your school.
Believe yourself, trust yourself, and mean yourself.
Because, we need you.
Please, believe yourself!
VIDEO 2
WHAT YOU KNOW ABOUT MATH
There's a lot of existing subject mater is which is the included in coverage of mathematics lesson. From the video, only mentioning just some of items or matter any kind of learned in mathematics study. There are some significant figure in math, that is the equals sign, the of plus sign, the multiplication sign, and the division sign. In mathematics there'is also ln open bracket x close bracket of is so-called number natural. then limit x come near do not till. And there's also how to make moderate a function. Others there's also lesson of trigonometry etc., there's also form of the graph of sine, and graph of exponential function.
VIDEO 3
SOLVING PROBLEM GRAPH
Let he function f be defined by f open bracket x close bracket equals x plus 1, if 2 times f open bracket p close bracket equals 20, what is the value of f open bracket 3 times p close bracket.
What is f when x equals 3 times p?
f open bracket x close bracket equals x plus 1, equivalent 2 times f open bracket p close bracket equals 20, equivalent f open bracket p close bracket equals p plus 1, equivalent p plus 1 equals 10, equivalent p equals 9. Insert p equals 9 into x equals 3 times p, so, x equals 3 times 9, x equals 27. f open bracket 27 close bracket equals x plus 1, f open bracket 27 close bracket equals 27 plus 1, f open bracket 27 close bracket equals 28.
In the x times y co-ordinate plane, the graph of x equals y to the power of 2 minus 4 intersects line l at open bracket o comma p close bracket and open bracket s comma t close bracket.
x equals y to the power of 2 minus 4,
line l : m equals open bracket y2 minus y1 close bracket over open bracket x2 minus x1 close bracket equals open bracket t minus p close bracket over open bracket s minus o close bracket.
make the assistive dot, if x equals o hence y equals p, and if x equals s hence y equals t.
hence its graph can be made.
VIDEO 4
PROPERTIES OF LOGARITHMS
log base b of x equals y equivalent b to the power of y equals x
notation : log base 10 of x equals log x; log base e of x equals ln x, ln x is natural logarithm.
Example :
1. log base 10 of 100 equals x, value of x is …
log base 10 of 100 equals x equivalent 10 to the power of x equals 100, equivalent 10 to the power 2 equals 100, so, the value of x is 2.
2. log base 2 of x equals 3, value of x is …
log base 2 of x equals 3 equivalent 2 to the power 3 equals x, equivalent 8 equals x, so, the value of x is 8.
3. log base 7 of open bracket 1 over 49 close bracket equals x, value of x is …
log base 7 of 1 over 49 equals x, equivalent 7 to the power of x equals 1 over 49, equivalent 7 to the power of x equals 1 over open bracket 7 to the power 2 close bracket, equivalent 7 to the power of x equals 7 to the power of negative 2, so, the value of x is negative 2.
log base b of m times n equals log base b of m base b plus log n
log base b of m over n equals log base b of m minus log base b of n
log base b of x to the power of n equals n times log base b of x
expand :
log base 3of open bracket x to the power of 2 times open bracket y plus 1 close bracket over z to the power of 3 close bracket,
equivalent log base 3 of open bracket x to the power of 2 times open bracket y plus 1 close bracket close bracket minus log base 3 of 2 to the power of 3,
equivalent log base 3 of x to the power 2 plus log base 3 of open bracket y plus 1 close bracket minus log base 3 of z to the power of 3.
VIDEO 5
GRAPH OF A RATIONAL FUNCTION
can have discontinuities
has a polynomial in the dominator
off limit
f open bracket x close bracket equals open bracket x plus 2 close bracket over open bracket x minus close bracket, when x equals 1, hence f open bracket x close bracket equals 3 over 0, that’s no good, bad, bad, bad.
Bad choice, break in function graph, insert x equals 0, hence f open bracket 0 close bracket equals open bracket 0 plus 2 close bracket over open bracket 0 minus 1 close bracket (rational). Insert x equals 1, hence f open bracket 1 close bracket equals open bracket 1 plus 2 close bracket over open bracket 1 minus 1 close bracket, equivalent 3 over 0(no rational).
Break discontinuity
Rational function don’t always work this way
Rational function denominator can be zero
Polynomial (smooth unbroken curve)
There is no value for the function
Break in the graph
Missing point on the graph
y equals x to the power of 2 minus x minus 6 in bracket over x minus 3, if we insert x equals 3, hence the graph is not possible, not feasible, and not allowed. This is missing point syndrome.
to overcome it, factor top and bottom
y equals open bracket x minus 3 close bracket times open bracket x plus 2 close bracket in square bracket over open bracket x minus 3 close bracket, so that y equals x plus 2, so that if we insert x equals 3, y equals 3 plus 2 equals 5.
VIDEO 6
TRIGONOMETRY
There is a special triangle with angle is phi. To look for the sine, cosines or tangent from a the special triangle can use the way of knocking by heart its formula easier, that is soh cah toa; by o is opposite, that is length mark with lines in a the triangle which ahead time angle is phi, h is hypotenuse, that is hypotenuse from a the special triangle, and a is adjacent, that is side residing in side from angle phi. Intend the soh is sine of phi equals opposite over hypotenuse. And intend from cah is cosines equals adjacent over hypotenuse. And intend from toa is tangent equals opposite over adjacent.
example:
There is a triangle with the angle phi and opposite is equals 4, hypotenuse equals 5, and adjacent equals 3. Look for the value from sin phi, cosines phi, and tangent phi.
Soh; sinus phi equals opposite over hypotenuse, sinus phi equals 4 over 5.
Cah; cosines phi equals adjacent over hypotenuse, cosines phi equals 3 over 5.
Toa; tangent phi equals opposite over adjacent, tangent phi equals 4 over 3.
DO YOU BELIEVE
After seeing video entitling do you believe, I embrace some of words said by the remarkable speaker is which still occupy the class of fifth elementary school
Do you believe?
I can do anything, dream anything, and become anything.
I can do anything, dream anything, and become anything because you believe me.
Let’s me ask you a question that’s easy.
Do you believe on my classmate.
Do you believe that every singles can graduate where classmate, school, and collage.
You better, because next week, I join with your school.
Believe yourself, trust yourself, and mean yourself.
Because, we need you.
Please, believe yourself!
VIDEO 2
WHAT YOU KNOW ABOUT MATH
There's a lot of existing subject mater is which is the included in coverage of mathematics lesson. From the video, only mentioning just some of items or matter any kind of learned in mathematics study. There are some significant figure in math, that is the equals sign, the of plus sign, the multiplication sign, and the division sign. In mathematics there'is also ln open bracket x close bracket of is so-called number natural. then limit x come near do not till. And there's also how to make moderate a function. Others there's also lesson of trigonometry etc., there's also form of the graph of sine, and graph of exponential function.
VIDEO 3
SOLVING PROBLEM GRAPH
Let he function f be defined by f open bracket x close bracket equals x plus 1, if 2 times f open bracket p close bracket equals 20, what is the value of f open bracket 3 times p close bracket.
What is f when x equals 3 times p?
f open bracket x close bracket equals x plus 1, equivalent 2 times f open bracket p close bracket equals 20, equivalent f open bracket p close bracket equals p plus 1, equivalent p plus 1 equals 10, equivalent p equals 9. Insert p equals 9 into x equals 3 times p, so, x equals 3 times 9, x equals 27. f open bracket 27 close bracket equals x plus 1, f open bracket 27 close bracket equals 27 plus 1, f open bracket 27 close bracket equals 28.
In the x times y co-ordinate plane, the graph of x equals y to the power of 2 minus 4 intersects line l at open bracket o comma p close bracket and open bracket s comma t close bracket.
x equals y to the power of 2 minus 4,
line l : m equals open bracket y2 minus y1 close bracket over open bracket x2 minus x1 close bracket equals open bracket t minus p close bracket over open bracket s minus o close bracket.
make the assistive dot, if x equals o hence y equals p, and if x equals s hence y equals t.
hence its graph can be made.
VIDEO 4
PROPERTIES OF LOGARITHMS
log base b of x equals y equivalent b to the power of y equals x
notation : log base 10 of x equals log x; log base e of x equals ln x, ln x is natural logarithm.
Example :
1. log base 10 of 100 equals x, value of x is …
log base 10 of 100 equals x equivalent 10 to the power of x equals 100, equivalent 10 to the power 2 equals 100, so, the value of x is 2.
2. log base 2 of x equals 3, value of x is …
log base 2 of x equals 3 equivalent 2 to the power 3 equals x, equivalent 8 equals x, so, the value of x is 8.
3. log base 7 of open bracket 1 over 49 close bracket equals x, value of x is …
log base 7 of 1 over 49 equals x, equivalent 7 to the power of x equals 1 over 49, equivalent 7 to the power of x equals 1 over open bracket 7 to the power 2 close bracket, equivalent 7 to the power of x equals 7 to the power of negative 2, so, the value of x is negative 2.
log base b of m times n equals log base b of m base b plus log n
log base b of m over n equals log base b of m minus log base b of n
log base b of x to the power of n equals n times log base b of x
expand :
log base 3of open bracket x to the power of 2 times open bracket y plus 1 close bracket over z to the power of 3 close bracket,
equivalent log base 3 of open bracket x to the power of 2 times open bracket y plus 1 close bracket close bracket minus log base 3 of 2 to the power of 3,
equivalent log base 3 of x to the power 2 plus log base 3 of open bracket y plus 1 close bracket minus log base 3 of z to the power of 3.
VIDEO 5
GRAPH OF A RATIONAL FUNCTION
can have discontinuities
has a polynomial in the dominator
off limit
f open bracket x close bracket equals open bracket x plus 2 close bracket over open bracket x minus close bracket, when x equals 1, hence f open bracket x close bracket equals 3 over 0, that’s no good, bad, bad, bad.
Bad choice, break in function graph, insert x equals 0, hence f open bracket 0 close bracket equals open bracket 0 plus 2 close bracket over open bracket 0 minus 1 close bracket (rational). Insert x equals 1, hence f open bracket 1 close bracket equals open bracket 1 plus 2 close bracket over open bracket 1 minus 1 close bracket, equivalent 3 over 0(no rational).
Break discontinuity
Rational function don’t always work this way
Rational function denominator can be zero
Polynomial (smooth unbroken curve)
There is no value for the function
Break in the graph
Missing point on the graph
y equals x to the power of 2 minus x minus 6 in bracket over x minus 3, if we insert x equals 3, hence the graph is not possible, not feasible, and not allowed. This is missing point syndrome.
to overcome it, factor top and bottom
y equals open bracket x minus 3 close bracket times open bracket x plus 2 close bracket in square bracket over open bracket x minus 3 close bracket, so that y equals x plus 2, so that if we insert x equals 3, y equals 3 plus 2 equals 5.
VIDEO 6
TRIGONOMETRY
There is a special triangle with angle is phi. To look for the sine, cosines or tangent from a the special triangle can use the way of knocking by heart its formula easier, that is soh cah toa; by o is opposite, that is length mark with lines in a the triangle which ahead time angle is phi, h is hypotenuse, that is hypotenuse from a the special triangle, and a is adjacent, that is side residing in side from angle phi. Intend the soh is sine of phi equals opposite over hypotenuse. And intend from cah is cosines equals adjacent over hypotenuse. And intend from toa is tangent equals opposite over adjacent.
example:
There is a triangle with the angle phi and opposite is equals 4, hypotenuse equals 5, and adjacent equals 3. Look for the value from sin phi, cosines phi, and tangent phi.
Soh; sinus phi equals opposite over hypotenuse, sinus phi equals 4 over 5.
Cah; cosines phi equals adjacent over hypotenuse, cosines phi equals 3 over 5.
Toa; tangent phi equals opposite over adjacent, tangent phi equals 4 over 3.
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