Logarithms are important throughout mathematics and science.
Establishing some basic properties or relations of Logarithms helps
us for faster computation of mathematical problems.
Some of the basic properties of Logarithms and proof of them are given below, they are also called theorems of logarithm.
1> For any positive x ,y and a:
Proof:
Suppose:
then, and
so,
thus
2> For any positive a
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The Logarithm.
If “x” is a number and,
Then , “y” is known as the logarithm of “x” to the base “a”.
For example:
:-The logarithm of 16 to the base 2 is 4 which can be shown as:
:-The logarithm of 8 to the base 2 is 3 which can be shown as:
The logarithmic Function:
The function in which the relationbetween input(let x) and and output(let y)
is gi
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Exponential Function:
Exponential function is the function which is defined by the following formula:
y=f(x)=ax
Where , a is a constant great than 0 and a , x both are real numbers.
In any exponential function defined by formula y=f(x)=ax,
“a” is said to be the base of exponential function “f” and “x” is the exponent of “a”.
Some Examples of Exponent
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Exponent or Index:
When a number (let a) is multiplied by itself multiple time(let n times) then we represent this case by adding “n” superscription to “a” as given below:
an
When , a number is repeatedly multiplied by itself the number of times the number is multiplied by itself is called the exponent the the number which is being multiplied is called base or index.
For example:
In an
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Trigonometric Functions:
If we place an angle in standard position or at origin and draw a circle with center at origin such that the circle will intersect terminal arm of the angle , As shown in following figure:
Where “x” is the x-coordinate of the point “P”, “y” is the y-coordinate of point “P” and “r” the radius of circle.
Then for
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Angle:
IF a line is rotated on one of it’s end point without changing the position of it’s another end point or it is rotated on one of it’s end point the the configuration formed by initial arm , terminal arm and vertex or common end point is known as Angle.
An angle always has following things:
An initial arm.
A terminal arm. and
A vertex.
As shown in figure below:
An angle
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The following types of functions are known as algebraic function:
Algebraic functions are also known as Polynomial functions.
a. The Identity Function:
The function defined by y=f(x)=x is known as identity function.
For example:
If I:R→R is a function defined by f(x)=x=y then the function “I” is a identity function.
We can show a identity function in graph as:
b. The Constant
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What is Composite Function?
Let f:A→B and g:B→C be any two functions , we can represent this two functions in a diagram as following.
What is happening here is the output of function “f” is again processed by another function “g” to give a new output.
If we make a new function (let the new function be “h”) which maps input of the function “fR
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Inverse of a Function:
Let “f” be a Function from set A to B or , elements of set A are changed or outputted to elements of set B if they are processed through or inputed in the function “f”.
or f:A→B
Then , inverse of the function “f” is a new rule or function (Let the new function be function “g”) , such that if we input the output of fun
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According to the nature shown by a Function it can be classified into different types.
They are as following:
1> One-To-One function.
A Function is said to be a One-To-One Function if every element of Domain of the function have its own and unique element in Range of the Function.
A function from set A to set B is said to be a One-To-One Function if no two or more elements of set A have
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