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.
Inverse FunctionBack to Top
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
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 LogarithmsBack to Top
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 FunctionBack to Top
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 EquationsBack to Top
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)
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 EquationsBack to Top
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.
Real ExponentsBack to Top
When we add or subtract the bases with real powers then we just have to add or subtract their coefficients (having same base). When multiplying numbers with real exponents’ only powers are added and similarly in the case of division powers are subtracted. This is what we can do when bases are same. In case of fraction we need to take the LCM (least common factor) to resolve our answer. For example consider the following opeReal Exponents
exponent definitionrations with real exponents:
10 4 / 5 + 10 2 / 5 + 10 2 / 5 + 2 (10 (4) / 5) = 3 (104 / 5) + 2 (102 / 5)
10 4 / 5 + 10 2 / 5 - 10 2 / 5 - 2 (10 (4) / 5) = - (104 / 5))
10 4 / 5 * 10 2 / 5 = 10 (4 + 2) / 5 = 106 / 5
10 4 / 5 / 10 2 / 5 = 10 (4 - 2) / 5 = 102 / 5.