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Logarithm and inverse function Table of contents Sl.No Topics Pages 1 Acknowledgement 2 2 Introduction 3 3 Logarithm 4-5 4 Logarithm identities 6-8 5 Logarithm function 9-16 6 Inverse function 17-28 7 Conclusion 29 8 References 30 Submitted by Tashi Yangchen (BE 102131) Page 1 Logarithm and inverse function Acknowledgement In this process of doing this assignment of logarithm and inverse function, I have got information from different sources like internet, number of books from library, and from my friends. Without above mentioned help, my assignment would not be in this form. So for that I would like to give my heartfelt gratitude to them individually. Lastly I would like to thank sir for giving assignment on logarithm and inverse function which gave me lots of information and ideas. Submitted by Tashi Yangchen (BE 102131) Page 2 Logarithm and inverse function Introduction In this modern world there should be mathematic background. Without mathematic background, it would be like living in the darkness because we need math in our day to day life for example if we want to buy vegetables we need mathematic background to calculate. So, where ever we go we need mathematic background, it is very important for all of us to know mathematic. There will be vast difference people having math background and people without having math background. Logarithm is one of topic under math. The logarithm of a number y with respect to a number b is the power to which b has to be raised in order to give y. In symbols, the logarithm is the number x satisfying the following equation: bx = y. The logarithmic function is defined as the inverse of the exponential function. For B > 0 and B not equal to 1, y = Log Bx is equivalent to x = B y. The inverse function is defined as the reverse process of the original function in reverse order. A relationship between two numbers is increase in the value of one number result in a decrease in the value of the other number. The inverse relation of a binary relation which the relation that occurs when you replace the order of the elements in the relation. For example, the inverse of the relation 'parent of’' is the relation 'child of'’.The analogy notation comes under an inverse function. Submitted by Tashi Yangchen (BE 102131) Page 3 Logarithm and inverse function Logarithm In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 3 is the power to which ten must be raised to produce 1000: 103 = 1000, so log101000 = 3. Only positive real numbers have real number logarithms; negative and complex numbers have complex logarithms. The logarithm of x to the base b is written logb(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y, The bases used most often are 10 for the common logarithm, e for the natural logarithm, and 2 for the binary logarithm. An important feature of logarithms is that they reduce multiplication to addition, by the formula: That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers. Similarly, logarithms reduce division to subtraction by the formula: That is, the logarithm of the quotient of two numbers is the difference between the logarithms of those numbers. Submitted by Tashi Yangchen (BE 102131) Page 4 Logarithm and inverse function Integral representation of the natural logarithm The natural logarithm of t is the shaded area underneath the graph of the function f(x) = 1/x (reciprocal of x). When b = e (Euler's number), the natural logarithm ln(t) = loge(t) can be shown to satisfy the following identity: Submitted by Tashi Yangchen (BE 102131) Page 5 Logarithm and inverse function Logarithm identities 1. Logarithm of a product The logarithm of a product is the sum of the two logarithms. That is, for any two positive real numbers x and y, and a given positive base b, the following identity holds: logb(x · y) = logb(x) + logb(y). For example, log3(9 · 27) = log3(243) = 5, since 35 = 243. On the other hand, the sum of log3(9) = 2 and log3(27) = 3 also equals 5. In general, that identity is derived from the relation of powers and multiplication: bs · bt = bs + t. Indeed, with the particular values s = logb(x) and t = logb(y), the preceding equality implies logb(bs · bt) = logb(bs + t) = s + t = logb(bs) + logb(bt). .2. Logarithm of a power The logarithm of the p-th power of a number x is p times the logarithm of that number. In symbols: logb(xp) = p logb(x). As an example, Submitted by Tashi Yangchen (BE 102131) Page 6 Logarithm and inverse function log2(64) = log2(43) = 3 · log2(4) = 3 · 2 = 6. This formula can be proven as follows: the logarithm of x is the number to which the base b has to be raised in order to yield x. In other words, the following identity holds: x = blogb(x). Raising both sides of the equation to the p-th power (exponentiation) shows xp = (blogb(x))p = bp · logb(x). (At this point, the identity (de)f = de · f was used, where d, e and f are positive real numbers.) Thus, the logb of the left hand side, logb(xp), and of the right hand side, p · logb(x), agree. The sought formula is proven. Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. For example, Submitted by Tashi Yangchen (BE 102131) Page 7 Logarithm and inverse function 2. Change of base The third important rule for calculating logarithms is the following formula calculating the logarithm of a fixed number x to one base in terms of the one to another base: This is shown as follows: the left hand side of the above is the unique number a such that ba = x. Therefore logk(x) = logk(ba) = a · logk(b). The general restriction b ≠ 1 implies logkb ≠ 0, since b0 = 1. Thus, dividing the preceding equation by logkb shows the above formula. One way of viewing the change-of-base formula is to say that in the expression the two bs "cancel", leaving logk x. Submitted by Tashi Yangchen (BE 102131) Page 8 Logarithm and inverse function LOGRATHMIC FUNCTION Definitions of Logarithmic Function The logarithmic function is defined as the inverse of the exponential function. For B > 0 and B not equal to 1, y = Log Bx is equivalent to x = B y. Note: The logarithm to the base e is written ln(x). Logarithm as a function The graph of the logarithm function logb(x) (green) is obtained by reflecting the one of the function bx (red) at the diagonal line (x = y). In the more highbrow language of calculus, the above proof amounts to saying that the function f(x) = bx is strictly increasing, takes arbitrarily big and small positive values and is continuous (intuitively, the function does not "jump": the graph can be drawn without lifting the pen). These Submitted by Tashi Yangchen (BE 102131) Page 9 Logarithm and inverse function properties, together with the intermediate value theorem of elementary calculus ensure that there is indeed exactly one solution x to the equation f(x) = bx = y, for any given positive y. The expression logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) refers to a function of the form logb(x) in which the base b is fixed and x is variable, thus yielding a function that assigns to any x its logarithm logb(x). The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function. The above definition of logarithms was done indirectly by means of the exponential function. A compact way of rephrasing that definition is to say that the base-b logarithm function is the inverse function of the exponential function bx: a point (t, u = b(t)) on the graph of the exponential function yields a point (u, t = logbu) on the graph of the logarithm and vice versa. Geometrically, this corresponds to the statement that the points correspond one to another upon reflecting them at the diagonal line x = y. Using this relation to the exponential function, calculus tells that the logarithm function is continuous (it does not "jump", i.e., the logarithm of x changes only little when x varies only little). What is more, it is differentiable (intuitively, this means that the graph of logb(x) has no sharp "corners"). Submitted by Tashi Yangchen (BE 102131) Page 10 Logarithm and inverse function Graphing Logarithmic Functions y = sqrt(x) y = log2(x) y = 2x comparison y = log2(x) of the two graphs, showing the inversion line in red . Submitted by Tashi Yangchen (BE 102131) Page 11 Logarithm and inverse function Exponential functions The exponential function with positive base b > 1 is the function y = bx. It is defined for every real number x. Here is its graph: There are two important things to note: • The y-intercept is at (0, 1). For, b0 = 1. • The negative x-axis is a horizontal asymptote. For, when x is a large negative number -- e.g. b−10,000 -- then y is a very small positive number. Example 1. Solve this equation for x : 5x + 1 = 625 Solution; To "release" x + 1 from the exponent, take the inverse function -- the logarithm with base 5 -- of both sides. Equivalently, write the logarithmic form log55>x + 1 Submitted by Tashi Yangchen (BE 102131) = log5625 Page 12 Logarithm and inverse function x+1 = log5625 x+1 = 4 x = 3. 2. Solve for x : 2x − 4 = 3x Solution; We may take the log of both sides either with the base 2 or the base 3. Let us use base 2: log22x − 4 = log23x x−4 = x log23, x − x log23 = 4 x(1 − log23) = 4 x = 4 1 − log23 log23 is some number. The equation is solved. Submitted by Tashi Yangchen (BE 102131) Page 13 Logarithm and inverse function 3. Solve for x: log5(2x + 3) = 3 Solution; To "free" the argument of the logarithm, take the inverse function -- 5x -- of both sides. That is, let each side be the exponent with base 5. Equivalently, write the exponential form. 2x + 3 = 53 2x = 125 − 3 2x = 122 x = 61 4. Solve for x : log4(3x − 5) = 0 5. Solve for x: log (2x + 1) = log 11 Solution. If we let each side be the exponent with 10 as the base, then according to the inverse relations: 2x + 1 = 11. That implies x = 5. 6. Use the laws of logarithms to write the following as one logarithm. log x + log y − 2 log z Solution. log x + log y − 2 log z = log xy − log z² Submitted by Tashi Yangchen (BE 102131) Page 14 Logarithm and inverse function = log xy z² 7. Write as one logarithm: k log x + m log y − n log z 8. According to this rule, n = logbbn, we can write any number as a logarithm in any base. For example, 7 = log227 5.9 = log335.9 t = ln et 3 = log 1000 9. Evaluate eln x Solution: Let y = eln x Taking ln of both sides, we have: ln y = ln x ln e = ln x Submitted by Tashi Yangchen (BE 102131) Page 15 Logarithm and inverse function This means that y = x. Or, eln x = x 10. Express 82 = 64 in logarithmic form. Ans. log864 = 2 Submitted by Tashi Yangchen (BE 102131) Page 16 Logarithm and inverse function INVERSE FUNCTION The inverse of a function is the set of ordered pairs obtained by interchanging the first and second elements of each pair in the original function. Notation: If f is a given function, then f -1 denotes the inverse of f. (If the original function is a one-to-one function, the inverse will also be a function.) If a function is composed with its inverse, More specifically: the result is the starting value. Think of it as the function and the inverse undoing one another when composed. Consider the simple function f (x) = {(1,2), (3,4), (5,6)} and its inverse f -1(x) = {(2,1), (4,3), (6,5)} The answer is the starting value If ƒ is a function from a set A to a set B, then an inverse function for ƒ is a function from B to A, with the property that a round trip (a composition) from A to B to A (or from B to A to B) returns each element of the initial set to itself. Thus, if an input x into the function ƒ produces an output y, then inputting y into the inverse function produces the output x, and vice versa. A function ƒ and its inverse ƒ–1. Because ƒ maps a to 3, the inverse ƒ–1 maps 3 back to a. Submitted by Tashi Yangchen (BE 102131) Page 17 Logarithm and inverse function Examples: 1. Find the inverse of the function Answer: Remember: Set = Swap y. the variables. Solve for y. 2. Find the inverse of the function Answer: Remember; Set = y. Swap Eliminate the the fraction by multiplying variables. each side by y. Get the y's on one side of the equal sign by subtracting y from each side. Isolate the y by factoring out the y. Solve for y. Submitted by Tashi Yangchen (BE 102131) Page 18 Logarithm and inverse function 3. Find the inverse functions of f given by f(x) = 2x + 3 Solution: Write the function as an equation. y = 2x + 3 Solve for x. x = (y - 3)/2 Now write f-1(y) as follows; f -1(y) = (y - 3)/2 or f -1(x) = (x - 3)/2 4. Find the inverse functions of f given by f(x) = (x - 3)2, if x >= 3 Solution Write the functions as an equation. y = (x - 3)2 Solve for x, two solutions; x=3+y x=3-y Submitted by Tashi Yangchen (BE 102131) Page 19 Logarithm and inverse function The first solution is selected since x >= 3, write f-1(y) as follows. f -1(y) = 3 + y or f -1(x) = 3 + x f(f -1(x))=((3+ x)-3)2 =(x)2 =x f -1(f(x))=3+ (x-3)2 =3+|x-3| (since x >= 3, x-3 >= 0, |x-3| = x-3) =3+(x-3) =x 5. Find the inverse functions of f given by f(x) = (x + 1)/(x - 2) Solution to example 3: Write the functions as an equation. y = (x + 1) / (x - 2) Multiply both sides of the above equation by x - 2 and simplify. y (x - 2) = x + 1 Submitted by Tashi Yangchen (BE 102131) Page 20 Logarithm and inverse function Multiply and group. y x - 2y = x + 1 y x - x = 2y + 1 Factor x on the left side and solve x(y - 1) = 1 + 2y x = (1 + 2y) / (y - 1) Change x to y and y to x y = (1 + 2x) / (x - 1) The inverse of functions f given above is f -1(x) = (1 + 2x) / (x - 1) 6. Find f-1(x) of 3x + 1 Solution: The equation is y = 3x + 1. Interchange x and y. x = 3y + 1 Solve for y. x - 1 = 3y (x - 1)/3 = y f-1(x) = (x - 1)/3 Submitted by Tashi Yangchen (BE 102131) Page 21 Logarithm and inverse function Graph; The graph of an inverse is the reflection of the original graph over the identity line, y = x. It may be necessary to restrict the domain on certain functions to guarantee that the inverse is also a function. Example: Consider the straight line, y = 2x + 3, as the original function. It is drawn in blue. If reflected over the identity line, y = x, the original function becomes the red dotted graph. The new red graph is also a straight line and passes the vertical line test for functions. The inverse of y = 2x + 3 is also a function. Not all graphs produce an inverse which is also a function. Let ƒ be a function whose domain is the set X, and whose codomain is the set Y. Then, if it exists, the inverse of ƒ is the function ƒ−1 with domain Y and codomain X, with the property: Stated otherwise, a function is invertible if and only if its inverse relation is a function, in which case the inverse relation is the inverse function. Not all functions have an inverse. For this rule to be applicable, each element y ∈ Y must correspond to exactly one element x ∈ X. This is generally stated as two conditions: Every corresponds to no more than one ; a function ƒ with this property is called one-to-one, or information-preserving. Submitted by Tashi Yangchen (BE 102131) Page 22 Logarithm and inverse function Every corresponds to at least one ; a function ƒ with this property is called onto function. A function with both of these properties is called a bijection, so the above is often stated as "a function is bijective if and only if it has an inverse function". In elementary mathematics, the domain is often assumed to be the real numbers, if not otherwise specified, and the co domain is assumed to be the image. Most functions encountered in elementary calculus do not have an inverse. If ƒ and ƒ−1 are inverses, then the graph of the function is the same as the graph of the equation This is identical to the equation y = ƒ(x) that defines the graph of ƒ, except that the roles of x and y have been reversed. Thus the graph of ƒ−1 can be obtained from the graph of ƒ by switching the positions of the x and y axes. This is equivalent to reflecting the graph across the line y = x. Submitted by Tashi Yangchen (BE 102131) Page 23 Logarithm and inverse function 7. Find the inverse of the function defined as . Solution: If this function has an inverse, we know that the composition of the function and its inverse equals x or . Recall that the original rule takes us from x to f(x) and the inverse rule takes us from f(x) back to x. The equation indicates that we use the rule associated with f(x) and as the argument. The equation can be written or . Solve for by isolating the term . Subtract 2 from both sides of the equation and we have . Take the cube root of both sides of the equation and we have Add 5 to both sides of the equation by 6 and we have The inverse function is the function Submitted by Tashi Yangchen (BE 102131) . . . Page 24 Logarithm and inverse function You can also graph the original function , the inverse function , and the line y = x. If you fold the graph paper over the line y = x, the graphs of the original function and the inverse function will be superimposed. 8. Find the inverse of the function defined as . Solution: If this function has an inverse, we know that the composition of the function and its inverse equals x or . Recall that the original rule takes us from x to f(x) and the inverse rule takes us from f(x) back to x. The equation indicates that we use the rule associated with f(x) with the argument . Submitted by Tashi Yangchen (BE 102131) Page 25 Logarithm and inverse function The equation Solve for can be written by isolating the term or . Multiply both sides of the equation by 3 and we have . Subtract 5 from both sides of the equation and we have Divide both sides of the equation by 2 and we have The inverse function is the function . . . . You can also graph the original function , the inverse function , and the line y = x. If you fold the graph paper over the line y = x, the graphs of the original function and the inverse function will be superimposed. Submitted by Tashi Yangchen (BE 102131) Page 26 Logarithm and inverse function 9. Find the inverse of the function defined as . Solution: If this function has an inverse, we know that the composition of the function and its inverse equals x or . Recall that the original rule takes us from x to f(x) and the inverse rule takes us from f(x) back to x. The equation indicates that we use the rule associated with f(x) with The equation can be written Solve for Add by isolating the term . or . . to both sides of the equation and we have . Simplify the right side of the above equation and we have Multiply both sides of the equation by 3 and we have . . Divide both sides of the equation by 7 and we have The inverse function is the function Submitted by Tashi Yangchen (BE 102131) . Page 27 Logarithm and inverse function You can also graph the original function , the inverse function , and the line y = x. If you fold the graph paper over the line y = x, the graphs of the original function and the inverse function will be superimposed. Submitted by Tashi Yangchen (BE 102131) Page 28 Logarithm and inverse function CONCLUSION By doing this assignment on inverse function and the logarithmic function we have got lots of information about inverse function and logarithmic functions. This assignment gives us much knowledge on these particular topics and now I have gained good concept on this two functions. By knowing only definition will not give good concept on this functions. But we need to go deep to have good concept about inverse function and logarithm function. Logarithms are found in applications such as compound interest, earthquake magnitudes, and the pH of solutions. A logarithmic function gives “exponent” of an expression in terms of a base, “a”, and a number, “x”. For example ay=x and f(x) = y= logax. Where the base “a” is positive real number, which excludes “1”. It is very interesting topic to learn. Therefore, logarithm and inverse function is the two subtopic which gives new skills and help us to solve problems in the better way. Submitted by Tashi Yangchen (BE 102131) Page 29 Logarithm and inverse function REFERENCES http://www.purplemath.com/modules/graphlog.htm http://en.wikipedia.org/wiki/Logarithm_function http://www.themathpage.com/aprecalc/logarithmic-exponential-functions.htm http://www.regentsprep.org/Regents/math/algtrig/ATP8/inverselesson.htm http://www.answers.com/topic/inverse-function Submitted by Tashi Yangchen (BE 102131) Page 30