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Exponents and Logarithms

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 Sub Topics In mathematics, logarithm is a function or a power function which is used to increases the number logarithmically. The Logarithm of 10000 to base 1000 is 4, because 10000 are 10 to the power 4. We can also write as: 10000 = 10 * 10 * 10 * 10; Suppose s = qt, then ‘t’ is the Logarithm of ‘s’ to the base ‘q’, and it is written as Logq s, so log10 (10000) = 4; Let’s see how to calculate exponents and logarithms Functions. To solve the exponential and logarithmic Functions we have to follow some steps which are given below: Step 1: To solve exponents and logarithms take a logarithm function. Step 2: Then put the exponent on both sides of the logarithm function. Step 3: As we know exponent value and logarithm function are same as it gets multiplied. Step 4: When we calculate the logarithm function we write the exponent value first and then log function. Step 5: If any value present inside the logarithm function, then it can be written according to the rule of logarithm function. Let’s talk about Exponential Function: The functions which are distinct in the power of ‘e’ is said to be exponential function. In mathematics, exponential function is used to illustrate the relationship, when we change the independent variable and it gives the same proportion change in the dependent variable. The value of exponential function is approximately 2.718281828. Suppose we have s = eS; Then the graph of this function will always be increasing as the value of ‘x’ increases. The graph of this function lies towards the x –axis. And the derivative of exponential function is its self. DeS = eS, dp, This is all about exponential and logarithmic function.

Inverse Function

Inverse function is the backtrack approach of a function. If there is a function with input ‘p’ and its giving the output as ‘q’ then the function which has input as ‘q’ and gives the output as ‘p’, that function will be called as the inverse function of a given function. When we perform a task and get an output, then if we treat that output as input and get the initial input value, this works as inverse function.
Function and the inverse of a function share a relation together. We can show that relation as:
f(g(p)) = g(f(p)) = q
In this if we inverse the function we will get the same result as q.
We can find the inverse just by the swapping of value p and q and we get a new relation which represents the inverse of a function. But we have to note that it is not necessary that an inverse is always a function.
Let us assume that we are given a function say f(p) and getting the output q then we will denote the inverse function with q input as f-1 (q). For understanding the inverse function in a better manner let us take another example. Assume a function f(p)=2p+6 then what will be its inverse function let’s take a look:
f(p) = 2p + 6
p ----> 2*p --- > 2*p + 6
This function will be processed in this manner. For calculating its inverse just does the opposite of each task or inverse of each task like replace addition with subtraction and multiplication with division.
f-1(q) will be
(q - 6)/2 < ----- q - 6 <---- q
So the inverse function f-1(q) of function f(p) is:
f-1(q) = (q - 6)/2
Mathematics approach:
step 1: f(p) = 2p + 6
step 2: Put q at the place of f(p)
q = 2p + 6
step 3: Take Integer value at left side
q - 6 = 2p
step 4: Divide left side by the coefficient of p
(q - 6) /2 = p
Step 5: p = (q - 6) / 2
Step 6: Put f-1(q) at the place of p.
f-1(q) = (q - 6) /2
This is the all about inverse of function.

Natural Logarithms

In mathematics, natural Logarithm is the Logarithm which has base ‘e’, where ‘e’ represents transcendental and irrational constants that is approximately equal to 2.718281828. The natural logarithm is basically written in the form of ln (s), and sometimes as loge (s), if the base of e is defined as simply log(s).
If the value of ln (7.389...) is defined as 2, because 2 = 7.389.... The natural log of ‘e’ itself (ln (e)) is 1 because e1 = e, while the natural logarithm of ‘1’ (ln (1)) is 0, since e0 = 1.
The natural logarithm can also be represented for any positive real number ‘a’ as the area along the given curve y = 1/x from 1 to ‘a’. The real valued function of any real function is considered as the natural log Functions which is the inverse or opposite function of the Exponential Function. Some identities are also defined for natural logarithm.
⇒ e ln (x) = x; if the value of x > 0;
We can also write it in logarithm form as:
⇒ ln (ex) = x.
Properties of natural logarithms:
1. ln (1) = 0;
2. ln (-1) = i⊼;
To find the logarithm we have to follow some steps, which are given as:
Step 1: To Define Logarithm first we have to take an exponential form value.
Step 2: If we change an exponential function into logarithm function, we have to take log on both sides of exponential function.
Step 3: After taking the log the subscript value becomes the base of logarithm and value which lies inside the logarithm is said as argument of the log.
This is all about natural logarithm definition.

Exponential Function

We generally study different type of Functions in the Math like the Trigonometric Functions which include the sine function, the cosine function, the Tangent function, the log function, the greatest Integer function, etc. One of such functions of the math which is the most common function is exponential function.
The exponential function in the context of the math is a function which is represented by ex in which ‘e’ denotes a number whose value is approximately equal to 2.718281828. This function is such type of the function which has the same derivative as itself that is the derivative is also equal to ex.
Exponential functions are generally utilized in the modeling of a kind of the relationship where a constant variation in the variable which is independent produces an equally proportional variation in the variable which is dependent that is it produces an equally proportional increase or the decrease when it is considered percent wise. This function is generally written in the form exp ( x ) and specially during the conditions when it is not practical to write the variable which is independent in the form of a superscript. The inverse of these functions are called as the log functions.
Now, let us discuss something about the graph of these functions. The graph of function y = ex is of sloping type which is sloping in the upward direction and also the graph of this function increases very rapidly with the increase in the value of the x. The graph of exponential exists always on the upper side of the x axis and gets close to the x axis for those values of the x which are negative. Hence we can say that the x axis can be called as the horizontal Asymptote of the function.

Solving Logarithm Equations

Logarithmic equations can be solved by applying regular algebraic operations. To solve these types of equations we should have good knowledge of how arithmetic operations are used with logs. Operations like addition, subtraction, multiplication, division, exponent etc. can also be found in solving Logarithm equations. General way of writing an algebraic equation is axn + bxn - 1+ cxn – 2 +……. + zx0. Logarithmic equations can also include exponents of any degree. These equations are an important opening concept in Algebra which can easily be solved using the relationship between logarithmic equations and exponentials. It becomes easy for us to solve these equations by using the logarithmic rules for solving multiplication and division. For instance, log (base c) (x) = a, then c a = x. Let us consider an example to understand solving Logarithm equations.

These equations can be used to determine value of an unknown variable which can be present as a base in the equation. Say for example we have to solve a logarithm equation given as log 2 (x + 1) + log 2 (x – 4x + 1) – log4 (x2) = 4, to solve this equation let us first convert complete equation in common base that is 2 in our case. We get,
log2 (x+1) + log2 (x – 4x + 1) – log2 (x2) / log2 4 = 4
or log2 (x+1) + log2 (x – 4x + 1) – log2 (x2) / log2 22 = 4
or log2 (x+1) + log2 (x – 4x + 1) – log2 (x2) / 2 = 4
Using the logarithmic rule for multiplication and division in common base:
Log x + Log y = Log xy and
Log x - Log y = Log (x / y)
We get,
or log2 [(x+1) * (– 3x + 1)] / [(x2) / 2] = 4
or -3x2+x-3x+1= 16 [(x2) / 2]
or -3x2-2x+1 =8x2
or 11x2+2-1=0 . This equation can be solved for x like we used to do earlier.

Solving Exponential Equations

An exponential equation has applications in many fields like finance, physics and the natural sciences and they can be solved using some predefined rules. An equation can be defined as a mathematical term which shows relationship between a dependent variable and an independent variable. In equations that we are going to deal with now has an independent variable appearing in exponent. These equations can be thought of like: y = ex. Here, 'y' is a dependent variable and have been expressed as an Exponential Function of independent variable 'x'. To understand solving exponential equations you must be comfortable with logarithmic operations. But it is not mandatory to solve exponential equations using log always. Some can be solved without logarithms by either predicting or altering base of exponent.

Let us see some examples of exponential equations. Say we have an equation given as:

2X2 + 17X + 21 = 0
This equation can be solved without taking log.
2X2 + 14X + 3X + 21 = 0
2X (X + 7) + 3 (X + 7) = 0
(2X + 3) (X + 7) = 0
X = -3 / 2 OR - 7
Similarly, consider an equation which can find the use of logarithmic operations: For example, 18 = 2 (3y) – 2.
Move all the terms without y in their exponent to one side of the equation. So, our equation now becomes: 20 = 2 (3y).
Taking log on both sides we get,
If you take log base 10, 20 = 2 (3y) becomes log10 (10 * 2) = log10 (2 (3y)).
Simplifying equation by using:
Log (ab) = b log (a) and log10 (a * b) = log10 (a) + log10 (b). We get,
Log10 (2) + Log10 (10) = log10 (2) + y (log10 3)
Or y = 1 / log3.