Learn online logarithms

LOGARITHMS, usually referred to as logs play a very important role in mathematics.They are very useful in areas like scientific computation and solving algebraic problems. Though they seem very complicated but they are very simple once you understand them.Let us look at the formal definition of logs.Logarithm of a number to a given base is the power to which the base needs to be raised in order to get the numeric result, that is, for any number x, base b and exponent y definition will be as follows:

`"If "x = b^y` , then `y = log_b (x)`

There are two types of logarithms, one is with the base '10' and the other with base as 'e'. Those with base e are known as natural logarithms and are denoted as 'ln' and with base 10 are written as log.Think of any number..........lets say 8,write down the possible ways of writing it.Ones which come to my mind are: 8*1, 10-2, 5+3, 4*2,,,,and so on.One way of writing it in log form is Log2256. We will learn how to compute these logarithms online in a simple way in this section.

Learn Algorithms Online : Laws of logarithms


Different rules or identities or laws of logarithms are as follows:

1) loga(xy)= logax+logay ( This is referred to as product law)

2) loga(x/y)= logax-logay (This is referred to as division law)

3) logaax=x ( since logaa=1)

4) logaxn=nlogax (Power rule)

5) blogbx=x

6) logbx= lobkx/logkb (Change of base formula)


Learn Logarithms Online : Examples


Examples are as follows:

1) log(12)= log(3*4)= log3+log4

2) log(25)=log(50/2)= log50-log2

3) log525=log552=2log55=2(1)=2

4) log12525= log525/log5125=log552/log553= 2log55/3log55= 2(1)/3(1)= 2/3


Examples are as follows:

1) log(12)= log(3*4)= log3+log4

2) log(25)=log(50/2)= log50-log2

3) log525=log552=2log55=2(1)=2

4) log12525= log525/log5125=log552/log553= 2log55/3log55= 2(1)/3(1)= 2/3

population variance formula

In the variance of a random variable or distribution is the expectation, or mean, of the deviation squared of that variable from its expected value or mean. Thus the variance is a measure of the amount of variation within the values of that variable, taking account of all possible values and their probabilities or weightings.

Population Variance formula explanation:


The population variance formula for discover the variance for the given population problem

Population Variance formula:

Here, N is the size of the population.

So X-an unbiased estimate of µ. The variance of the population

Where µ is the population mean. n values x1, ..., xn from the population, This is merely a individual case of the universal definition of variance introduced above, but controlled to finite populations.

In many functional situations, the true variance of a population is not known a priori and must be computed someway. When making with countless populations, this is generally impossible.

Most of the time it is not possible to obtain data for the entire population. For example, it is impossible to measure the weight of each male in a particular area to determine the average weight and variance for males of a particular area. In such cases, results for the population have to be estimated using samples.

Population variance formula for Example problems:


Example:

The hourly wages earned by a sample of five students are:

7, 5, 11, 8, 6.

Let a sample consist of n independent readings x1, x2...xn, drawn from a population which is not necessarily Gaussian. We know that the mean µ of our sample is given by

µ  =?X/N

µ = 37/5

µ = 7.40

For this problem the population variance

Formula = ?2 = ?(X-?)2/N

(X- µ)= 7-7.4=.4

(X- µ)= 5-7.4=-2.4

(X- µ)= 11-7.4=3.6

(X- µ)= 8-7.4=0.6

(X- µ)= 6-7.4=-1.4

We take squared on each (X- µ)

=.4 =0.16

=-2.4 =5.76

=3.6=12.96

=0.6=0.36

=-1.4=1.96

?2 = ?(X-?)2/N

?2=21.2/5

Population  variance   ?2=4.21

Solving Online Geometry Right Triangles

Learning online is one of the easiest ways to acquire knowledge of something. For the people who is not going to school or don’t have time to go to school and even for students, online learning is an interactive way of learning Triangle is a three sided polygon. There are many types of triangles. Here we are seeing about solution of right angled triangles. We can use the Pythagoras Theorem for solving a right angled triangle problem.Here we are going to study about how to solve the geometry right triangles problems and its example.

Pythagorean Theorem Calculator

Triangle ABC is right angled at C. So that AB is the hypotenuse and AC and BC are the sides of the right triangle, then the following relation holds true.

c2=a2+b2

Solving Online Geometry Right Triangles - Example Problems.

Example: 1

calculate  the length of the hypotenuse of a right angled triangle, given the lengths of the other two sides are 5 inches and 9 inches.



Solution:

We know that Pythagorean Theorem Formula,

a2 + b2=c2

Given:

a = 8

b = 11

c= to find

Substitute the a, b value in this equation,

(82 +112) =c2

Simplify the above we get

64 +121 = c2

185 = c2

Taking square root on both sides,

` sqrt(185)` = 10.60

Therefore the value of hypotenuse is equal to 13.60


Example: 2

Determine whether the given triangle is a right triangle.

Given:

Hypotenuse =10

Adjacent= 8

Opposite= 6

Formula:

c2=a2 + b2

Substitute a, b, c value

(1002) = (82+62)

Simplify the above we get,

100 = 64 +36

100 = 100

So the given triangle is a right angled triangle

Solving Online Geometry Right Triangles - Example: 3

Find the lengths of the hypotenuse of right angled triangle, given the lengths of the other two sides are 7 inches and 9 inches.

Solution:

We know that formula,

a 2 + b2 =c2

Given:

a = 7

b = 9

c = to find

Substitute the a, b value in this equation,

(72 +92) =c2

49 +81 = c2

130 = c2

Taking square root on both sides,

`sqrt(130)` = 11.4

Therefore the value of hypotenuse is 11.4

Solve Converse of Mid-point Theorem Problems

In this article we are going to solve converse of mid-point theorem problems. The straight line which is drawn through the mid point of one side of a triangle is parallel to another side bisects the third side. This is the converse of mid-point theorem. But the mid point theorem states that the line segments joining the mid points of any two sides of a triangle is parallel to the third side and equal to half of it.

Solve Converse of Mid-point Theorem Problems:

Given: In Triangle ABC, D is the midpoint of AB and DE is drawn parallel to BC.

To prove: AE = EC

Construction: Draw CF parallel to BA to meet DB proceed at F.




Proof : 

Statement	Reason
DB || BC BD || CF BCFD is a parallelogram Therefore BD = CF ... ( i ) BD = AD ... ( ii ) Therefore AD = CF In triangle ADE and CFE, we have AD = CF Angle ADE = Angle CFE Angle AED = Angle CEF Therefore triangle ADE = Triangle CFE Therefore AE = EC Therefore DE bisects AC Hence proved	Given By construction Both pairs of opposite sides are parallel Opposite sides of a parallelogram unequal D is the mid point of AB From ( i ) and ( ii ) Just proved Alternate to angle S Vertically opposite angles of S Angle Angle Side criterion C. P. T. C.

Example Problem - Solve Converse of Mid-point Theorem Problems:

ABC of a right angled triangle at B, Here the mid-point of AC is P.

Solve that PB = PA = ½


Solution for the  converse of mid-point theorem problems:

Given:

ABC of a right angled triangle at B, Here the mid-point of AC is P.

To prove:

PB = PA = `1/2` AC.

Construction:

Through P, draw a line parallel to BC, meeting AB at Q.

Proof:

AQP = ABC (corresponding angles)

AQP = 90 degree.

ABC = 90 degree

In D APQ and D BPQ

AQ = BQ

AQP = BQP = 90 degree

PQ = PQ

Therefore in triangle APQ = Triangle BPQ

PA = PB (corresponding parts of congruent triangles)

PA = PB = `1/2` AC

PA = `1/2` AC (given)

Area and Distance Calculator

Area is a number expressing the 2D size of a defined part of a surface, usually a region enclosed by a closed curve. Distance between (x1, y1) and (x2, y2) two points are called as distance.

Area of square = `a^2 `

Distance between two points = `sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]`

The area and distance calculator example problems and practice problems are given below.


Example Problems for Area and Distance Calculator:

Example problem 1:

Find the area of the square, whose side edge is  12.5meters

Solution:

Given:

Side (a) = 12.5 m

Area of square calculator is given below.



Area of square = a2

= 12.5 * 12.5

After simplify this, we get

Area of square = 156.25 square units

Example problem 2:

Find the area of the square, whose side length is 23.4 meters

Solution:

Given:

Side (a) = 23.4 m

Area of square calculator is given below:



Area of square = a2

= 23.4 * 23.4

After simplify this, we get

Area of square = 547.56 square units

Example problem 3:

Find the distance between two points A (4, 1) and B (6, 5)

Solution:

Distance between two points calculator is given below.



The distance between two points are (x1, y1) and (x2, y2)

Here, x1 = 4, x2 = 6, y1 = 1, y2 = 5

By using the following formula

Distance formula = `sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2]`

= `sqrt[(6 - 4)^2 + (5 - 1)^2]`

= `sqrt[(2)^2 + (4)^2]`

= `sqrt(4 + 16)`

After simplify this, we get

Distance = `sqrt(20) `

So, the distance between two points A (4, 1) and B (6, 5) is `sqrt(20)`

The above examples are helpful to study of area and distance calculator.


Practice Problems for Area and Distance Calculator:

Practice problem 1:

Find the area of the square, whose side length is 34.5 meters

Answer: Area of square =1190.25

Practice problem 2:

Find the distance between two points A (0.7, 22) and B (0.2, 1.8)

Answer: Distance between two points = `sqrt(408.28)`

Null Set Definition

A set is a collection of well defined objects. What do we mean by well defined objects? What is your favourite subject? Your answer may be math, science or history or geography. If you ask the same question to your friend, then she may say commerce. The answer differs from person to person. It is not well defined. If you ask the following question “Name the days in a week?” to anyone, then the answer will be universally same. It is well defined.

Introduction to Null Set Definition:

Consider the following sets.

The set of vowels in English alphabet = {a, e, i, o, u} --- 1

Set of numbers divisible by 2 = {2, 4, 6, 8, 10…} --- 2

The number of elements in a set is called cardinal number of the set. If A = {a, e, i, o, u}, then cardinal number of set A, denoted as n (A) = 5.

If we could write the cardinal number of the set then it is called finite set. Set 1 is a finite set. We could not count the number of elements in set 2. So it is called as infinite set.

Definition of null set:

A null set is a set whose cardinal number is zero. In other words, a null set is a set which has no elements. Null set is also called as empty set or void set. The empty set is denoted by the symbol {} or φ

Problems on Null Set:

Ex :   Let A = {x: 2< x <3, x is a whole number}. Then A is the empty set, because there is no natural number between 2 and 3.

Sol : Let B = {x: x² – 5 = 0 and x is a natural number}. Then B is an empty set because the equation x² – 5 = 0 is not satisfied by ant rational value of x.

Let C = {x: x² = 25, x is even}. Then C is an empty set, because the equation x² = 25 is not satisfied by any even value of x.

Solve for Y in Function Table

Function tables playing an important role in mathematics. Function is defined as the set of order pairs (x, y).  For example {(1,2) (4,3) }. A common rule is followed in the function that is y = 3x +1 (or)    y = x2. In this article we shall discuss about how to solve the function table with suitable example problems.


Example Problem on Function Table:

Solve for y in the function tale y = 3x +5.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = 3x + 5
0	y = 3 (0) + 5 = 5
1	y = 3(1) +5 = 8
2	y = 3(2) +5 = 11
-1	y = 3(-1) +5 = 2
-2	y = 3(-2) +5 = -1

Hence the order pair for the above function is (0, 5) (1, 8) (2, 11) (-1, 2)(-2, -1)

Example Problem on Function table:

Solve for y in the function tale y = x + 4.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = x + 4
0	y = (0) + 4 = 4
1	y = (1) +4 = 5
2	y = (2) +4 = 6
-1	y = (-1) +4 = 3
-2	y = (-2) +4 = 2

Hence the order pair for the above function is (0, 4) (1, 5) (2, 6) (-1, 3)(-2, 2)

Example Problem on Function Table:

Solve for y in the function tale y = x2 + 2.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = x2 + 2
0	y = (0)2 + 2 = 2
1	y = (1)2 +2 = 3
2	y = (2)2+2 = 6
-1	y = (-1)2 +2 = 3
-2	y = (-2)2 +2 = 6

Hence the order pair for the above function is (0, 2) (1, 3) (2, 6) (-1, 3)(-2, 6)

Example Problem on Function table:

Solve for y in the function tale y = 2x2 + 3.

Solution:

From the given function plug the different value of x and get the y value. In general x is input value and y is the output value.

Function Table:

x	y = 2x2 + 3
0	y = 2(0)2 + 3 = 3
1	y = 2(1)2 +3 = 5
2	y = 2(2)2+3 = 11
-1	y = 2(-1)2 +3 = 5
-2	y = 2(-2)2 +3 = 11

Hence the order pair for the above function is (0, 3) (1, 5) (2, 11) (-1, 5)(-2, 11)