Natural Logarithms Properties

Introduction to natural logarithm:The most important function-inverse pair in mathematics and science is the pair consisting of the natural logarithmfunction ln x and the exponential function ex. The key to understanding ex is ln x, so we introduce ln x first. The importance of logarithms came at first from the improvement they brought to arithmetic. The revolutionary properties of logarithms made possible the calculations of the great seventeenth-century advances in offshore navigation and celestial mechanics. Nowadays we do complicated arithmetic with calculators, but the properties of logarithms remain as important as ever.
natural Logarithm Function

The natural logarithm of a positive number x, written as ln x, is the value of an integral.Natural logarithm definition: The natural logarithmfunction

If x > 1, then ln x is the area under the curve y = 1/t from t = 1 to t = x. For 0 < x < 1, ln x gives the negative of the area under the curve from x to 1. The function is not defined for x = 0. We also have

Using x for everything would have us writing

with x meaning two different things. So, we change the variable of integration to t.
The graph of y = ln x and its relation to the function y = 1/x, x > 0. The graph of the logarithm rises above the x-axis as x moves from 1 to the right, and it falls below the axis as x moves from 1 to the left.

Properties of natural logarithms
The properties that made logarithms the single most important improvement in arithmetic before the advent of modern computers are listed below. The properties that made it possible to replace multiplication of positive numbers by addition and division of positive numbers by subtraction. They also made it possible to replace exponentiation by multiplication.
For any numbers a > 0 and x > 0
• The first property of Natural Logarithmsis Product rule , product of logarithm of x and y is equal to the sum of logarithm x and logarithm y: lnxy = ln x + ln y
• The second property of natural logarithmis quoitent rule, logarithm of division of x and y is equal to the difference of logarithm x and logarithm y:ln x/y = lnx – lny
• The third property of natural logarithm is Reciprocal rule:ln1/x = - ln x
• The forth property of natural logarithm is power rule: lnxn = n ln x.
Solving Natural Logarithms:  Let us understand the concept of solving natural logarithm with natural logarithm properties . Suppose we have to solve the natural logarithmwithout calculator ln e4. Now using the power property of natural logarithms, we have, 4 ln e. now simplifying using loge e = 1, we have the solution 4 . 1 = 4.

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Median in Mathematics

Median in mathematics is the middle value of the data, which separates the data into two equal halves; which means fifty percent of the numbers are above the median and fifty percent of the numbers are below the median. So, the Definition of Median can be written as a mathematical result that indicates that one half of the data or group is higher and one half lower. For instance, if there are 5 children in Smiths family of ages 5, 12, 7,9, 3, 15; how do we find the age of the middle child? We need to find the middle child first, so we need to put the ages in an increasing order. The order would be 3, 5, 7,9,12,15. From this ordered list, we can easily make out that 7 is the middle value and that is the Median of the data. So, we can conclude that the age of the middle child is 7 years.

If there is ‘n’ number of values in a given data; then the Median Formula is:
Median = (number of values in the data)/2= (n+1)/2
  Median Formula =(n+1)/2
While Calculating Median or Finding the Median of a given data, the steps are:
Step1. Arrange the list in the ascending order
Step2. Count the number of values in the list (whether even or odd)
Step3. If the number of values in the list is odd; then median is the value exactly in the middle of the list and if the number of values in the list is even; then the mean of the two middle values is the median

Let us Find the Median or Find Median of the given data.
 A marathon race was completed by 5 participants. How do you find the median of the race, given the data 3.6hrs 4.2hrs 3.2hrs 5.4hrs 4.8hrs?
Let us first arrange the data in the ascending order, 3.2, 3.6, 4.2, 4.8, 5.4
The number of values in the data, n=5. So, the third value 4.2hrs which is exactly in the middle is the median of the race.

The Jones Family drove through 6 states on their summer vacation. Find the median of gasoline price if the gasoline prices varied from state to state.  $3.29, $3.34, $3.27, $4.02, $3.60, $3.86
To find the median we need to arrange the data in increasing order.
$3.27,  $3.29,  $3.34,  $3.60,  $3.86,  $4.02
Here,  n=6 ; data has even number of values. So, median would be the mean of the two middle values. The two middle values are 3.34 and 3.60.
Median=(3.34+3.60)/2 = 3.47
The median of the gasoline rate is $3.47

Exponential and Logarithmic Functions

Exponential function: if a positive real number  other than 1  , then the function f(x) defined by f(x) = ax for all x belong to R is called as exponential function.the domain of this exponential function is R .It is evident from its graph that it is everywhere continuos.
Graph of exponential function

Graph of y= ax , where 0

Graph of y= ax, where a>1

The shape of the graph y= ax for any a>1 is essentially the same as that of y = 2x.The graph of y = ax for a typical base b>1 and is rising (when viewed from left  to right ).for a base b where 0

Example of exponential function: y= 2x , y=3x+3
Logarithm function:


Si nce the exponential function y= ax for a >0 , a not equal to1  is montonic  (its graph is either  always rising for all x always falling ) , it must have an inverse that is itself monotonic .the inverse function  is called the logarithm of x to the base a .
Definition of logarithm function : If a > 0 and a  not equal to 1, the logarithm x to the base b is the function y=   that satisfy ay=x , that is y=   mean ay=x. Graph of logarithm function 

 Here green line represents graph of y=ax and blue line represent y=   .
Because y = bx is a continuos  , increasing function that satisfies ax >0 for all x ,   must also be continuos and  increasing  , and its graph must lies entirely  to the right of y-axis.

Introduction to Polynomials

Monomial: Monomial is a constant or variable or product of variables or variables with exponents 0, 1, 2, 3….
Example: 
What is a Polynomial?  
Polynomial is an expression which includes constants, variables. There may be more than one variable and variables can have any real number as coefficients but variable exponents should be a whole number, that is, 0, 1, 2, 3…
Clearly, polynomial is an extension of monomial. Polynomial is an expression which involves one or more than one monomial which is separated by addition or subtraction. Each monomial in a polynomial is also called as a term.
Example: 

Polynomial is a general term it can use for all expressions which satisfies above conditions. More specifically, a polynomial with one term is called as a MONOMIAL, two terms is called as a BINOMIAL and three terms is called TRINOMIAL.

Adding and Subtracting Polynomials:
How To Add Polynomials:
Adding polynomials is similar to adding variables or constants, to add two or more polynomials we have to add like terms (or monomials) in those polynomials.
Example:

Subtracting polynomials:
Subtracting polynomials is similar to subtracting variables or constants, to subtract two or more polynomials we have to subtract like terms (or monomials) in those polynomials.
Example:

How to multiply polynomials?
Multiplying Polynomials is different from adding and subtracting polynomials, here each term in one polynomial should multiply with each term in another polynomial it has nothing to do with whether they are like terms or not.
Example:

Derivatives of Functions in Calculus

What are Derivatives?
Derivative is the measure of the rate of change at any given point on a curve. The rate of change of a function is the slope of a line.
The derivative of f(x) when the limit extends from h to 0 is given by the formula:

Derivative of Acceleration
Derivative of acceleration is the rate of change of acceleration with respect to time. It is referred to as ‘jerk’ in technical terms.
How to Find the Derivative?
We have general Derivative Formulas which help to find the derivative of the given function.
Let us consider the following examples:
Derivative Examples:

Prime Numbers

Prime numbers are positive whole numbers which are divisible by number 1 and itself.  Prime numbers have only two factors. The smallest prime number is 2. Now the question arises is 1 a prime number? No, 1 is not a prime number. So our next question will be why 1 is not a prime number? As per the definition of prime numbers, a prime number must have two factors but 1 only has one factor. It is the only positive integer with exactly one positive divisor.  Therefore it is not considered as a prime number.

Is 2 a prime number? 2 is a prime number as it is divisible by 1 and itself. It has two factors and as per the definition of a prime number, prime number must have two factors. Therefore, 2 is considered as a prime number and it is the only even number which is considered as prime. The other prime numbers are divisible by 2 also so they do not come under a category of a prime number.

Euclid proved that there are infinite prime numbers and there is always a prime greater than the largest known prime number. Many mathematicians tried finding out the largest prime number. The largest prime number found in 2008 was 2^43112609-1

Let us learn about relatively prime numbers. Two numbers that have only 1 as a common factor are called relatively prime numbers. Thus, any two prime numbers are relatively prime numbers. They are also called co prime numbers. However, composite numbers may also be co - prime to each other.

For example: - Numbers 3375 and 2744 are considered as relatively prime because:-
3375 = 1 X 3 X 3 X 3 X 5 X 5 X5
2744 = 1 X 2 X 2 X 2 X 7 X 7 X 7
3375 and 2744 have just 1 as a common factor so they are considered as relatively prime numbers.

Equivalent Fractions Learning

Learn about equivalent fraction here.

Two fractions are equivalent when the value of both the fractions are similar. First we need to simplify fraction and then compare the fractions and if the values are similar, they are equivalent fractions.

Steps to identify equivalent fractions are:
Step 1) Simplify the fractions
Step 2) Reduce to the simplest form
Step 3) Compare the fractions

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