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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)
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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.
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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.
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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:
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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,
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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,
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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.
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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
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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").
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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
.
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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
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= log5625
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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.
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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)
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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
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Logarithm and inverse function
This means that y = x. Or, eln x = x
10. Express 82 = 64 in logarithmic form.
Ans. log864 = 2
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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.
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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.
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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
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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
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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
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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.
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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.
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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)
.
.
.
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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
.
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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)
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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)
.
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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)
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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.
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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
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