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1 | 2118-2121 | e , sin–1 x ≠
,
2 2
−π π
, i e , x ≠ – 1, 1,
i |
1 | 2119-2122 | , sin–1 x ≠
,
2 2
−π π
, i e , x ≠ – 1, 1,
i e |
1 | 2120-2123 | e , x ≠ – 1, 1,
i e , x ∈ (– 1, 1) |
1 | 2121-2124 | , x ≠ – 1, 1,
i e , x ∈ (– 1, 1) To make this result a bit more attractive, we carry out the following manipulation |
1 | 2122-2125 | e , x ∈ (– 1, 1) To make this result a bit more attractive, we carry out the following manipulation Recall that for x ∈ (– 1, 1), sin (sin–1 x) = x and hence
cos2 y = 1 – (sin y)2 = 1 – (sin (sin–1 x))2 = 1 – x2
Also, since y ∈
,
2 2
π π
−
, cos y is positive and hence cos y =
2
1
x
−
Thus, for x ∈ (– 1, 1),
2
1
1
cos
1
dy
dx
y
x
=
=
−
2
1
1
x
−
2
1
1
x
−
−
2
1
1
x
+
f(x) sin–1 x cos-1 x tan-1x
Domain off (-1, 1) (-1, 1) R
f 1(x)
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
125
EXERCISE 5 |
1 | 2123-2126 | , x ∈ (– 1, 1) To make this result a bit more attractive, we carry out the following manipulation Recall that for x ∈ (– 1, 1), sin (sin–1 x) = x and hence
cos2 y = 1 – (sin y)2 = 1 – (sin (sin–1 x))2 = 1 – x2
Also, since y ∈
,
2 2
π π
−
, cos y is positive and hence cos y =
2
1
x
−
Thus, for x ∈ (– 1, 1),
2
1
1
cos
1
dy
dx
y
x
=
=
−
2
1
1
x
−
2
1
1
x
−
−
2
1
1
x
+
f(x) sin–1 x cos-1 x tan-1x
Domain off (-1, 1) (-1, 1) R
f 1(x)
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
125
EXERCISE 5 3
Find dy
dx in the following:
1 |
1 | 2124-2127 | To make this result a bit more attractive, we carry out the following manipulation Recall that for x ∈ (– 1, 1), sin (sin–1 x) = x and hence
cos2 y = 1 – (sin y)2 = 1 – (sin (sin–1 x))2 = 1 – x2
Also, since y ∈
,
2 2
π π
−
, cos y is positive and hence cos y =
2
1
x
−
Thus, for x ∈ (– 1, 1),
2
1
1
cos
1
dy
dx
y
x
=
=
−
2
1
1
x
−
2
1
1
x
−
−
2
1
1
x
+
f(x) sin–1 x cos-1 x tan-1x
Domain off (-1, 1) (-1, 1) R
f 1(x)
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
125
EXERCISE 5 3
Find dy
dx in the following:
1 2x + 3y = sin x
2 |
1 | 2125-2128 | Recall that for x ∈ (– 1, 1), sin (sin–1 x) = x and hence
cos2 y = 1 – (sin y)2 = 1 – (sin (sin–1 x))2 = 1 – x2
Also, since y ∈
,
2 2
π π
−
, cos y is positive and hence cos y =
2
1
x
−
Thus, for x ∈ (– 1, 1),
2
1
1
cos
1
dy
dx
y
x
=
=
−
2
1
1
x
−
2
1
1
x
−
−
2
1
1
x
+
f(x) sin–1 x cos-1 x tan-1x
Domain off (-1, 1) (-1, 1) R
f 1(x)
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
125
EXERCISE 5 3
Find dy
dx in the following:
1 2x + 3y = sin x
2 2x + 3y = sin y
3 |
1 | 2126-2129 | 3
Find dy
dx in the following:
1 2x + 3y = sin x
2 2x + 3y = sin y
3 ax + by2 = cos y
4 |
1 | 2127-2130 | 2x + 3y = sin x
2 2x + 3y = sin y
3 ax + by2 = cos y
4 xy + y2 = tan x + y
5 |
1 | 2128-2131 | 2x + 3y = sin y
3 ax + by2 = cos y
4 xy + y2 = tan x + y
5 x2 + xy + y2 = 100
6 |
1 | 2129-2132 | ax + by2 = cos y
4 xy + y2 = tan x + y
5 x2 + xy + y2 = 100
6 x3 + x2y + xy2 + y3 = 81
7 |
1 | 2130-2133 | xy + y2 = tan x + y
5 x2 + xy + y2 = 100
6 x3 + x2y + xy2 + y3 = 81
7 sin2 y + cos xy = κ
8 |
1 | 2131-2134 | x2 + xy + y2 = 100
6 x3 + x2y + xy2 + y3 = 81
7 sin2 y + cos xy = κ
8 sin2 x + cos2 y = 1
9 |
1 | 2132-2135 | x3 + x2y + xy2 + y3 = 81
7 sin2 y + cos xy = κ
8 sin2 x + cos2 y = 1
9 y = sin–1
2
2
1
x
x
+
10 |
1 | 2133-2136 | sin2 y + cos xy = κ
8 sin2 x + cos2 y = 1
9 y = sin–1
2
2
1
x
x
+
10 y = tan–1
3
2
3
,
1
3
x
x
x
−
−
1
1
3
3
x
−
<
<
11 |
1 | 2134-2137 | sin2 x + cos2 y = 1
9 y = sin–1
2
2
1
x
x
+
10 y = tan–1
3
2
3
,
1
3
x
x
x
−
−
1
1
3
3
x
−
<
<
11 2
1
2
1
,
cos
0
1
1
x
y
x
x
−
−
=
<
<
+
12 |
1 | 2135-2138 | y = sin–1
2
2
1
x
x
+
10 y = tan–1
3
2
3
,
1
3
x
x
x
−
−
1
1
3
3
x
−
<
<
11 2
1
2
1
,
cos
0
1
1
x
y
x
x
−
−
=
<
<
+
12 2
1
2
1
,
sin
0
1
1
x
y
x
x
−
−
=
<
<
+
13 |
1 | 2136-2139 | y = tan–1
3
2
3
,
1
3
x
x
x
−
−
1
1
3
3
x
−
<
<
11 2
1
2
1
,
cos
0
1
1
x
y
x
x
−
−
=
<
<
+
12 2
1
2
1
,
sin
0
1
1
x
y
x
x
−
−
=
<
<
+
13 1
22
,
cos
1
1
1
x
y
x
x
−
=
− <
<
+
14 |
1 | 2137-2140 | 2
1
2
1
,
cos
0
1
1
x
y
x
x
−
−
=
<
<
+
12 2
1
2
1
,
sin
0
1
1
x
y
x
x
−
−
=
<
<
+
13 1
22
,
cos
1
1
1
x
y
x
x
−
=
− <
<
+
14 (
)
1
2
1
1
,
sin
2
1
2
2
y
x
x
x
−
=
−
−
<
<
15 |
1 | 2138-2141 | 2
1
2
1
,
sin
0
1
1
x
y
x
x
−
−
=
<
<
+
13 1
22
,
cos
1
1
1
x
y
x
x
−
=
− <
<
+
14 (
)
1
2
1
1
,
sin
2
1
2
2
y
x
x
x
−
=
−
−
<
<
15 1
12
1
,
sec
0
2
1
2
y
x
x
−
=
<
<
−
5 |
1 | 2139-2142 | 1
22
,
cos
1
1
1
x
y
x
x
−
=
− <
<
+
14 (
)
1
2
1
1
,
sin
2
1
2
2
y
x
x
x
−
=
−
−
<
<
15 1
12
1
,
sec
0
2
1
2
y
x
x
−
=
<
<
−
5 4 Exponential and Logarithmic Functions
Till now we have learnt some aspects of different classes of functions like polynomial
functions, rational functions and trigonometric functions |
1 | 2140-2143 | (
)
1
2
1
1
,
sin
2
1
2
2
y
x
x
x
−
=
−
−
<
<
15 1
12
1
,
sec
0
2
1
2
y
x
x
−
=
<
<
−
5 4 Exponential and Logarithmic Functions
Till now we have learnt some aspects of different classes of functions like polynomial
functions, rational functions and trigonometric functions In this section, we shall
learn about a new class of (related) functions called exponential functions and logarithmic
functions |
1 | 2141-2144 | 1
12
1
,
sec
0
2
1
2
y
x
x
−
=
<
<
−
5 4 Exponential and Logarithmic Functions
Till now we have learnt some aspects of different classes of functions like polynomial
functions, rational functions and trigonometric functions In this section, we shall
learn about a new class of (related) functions called exponential functions and logarithmic
functions It needs to be emphasized that many statements made in this section are
motivational and precise proofs of these are well beyond the scope of this text |
1 | 2142-2145 | 4 Exponential and Logarithmic Functions
Till now we have learnt some aspects of different classes of functions like polynomial
functions, rational functions and trigonometric functions In this section, we shall
learn about a new class of (related) functions called exponential functions and logarithmic
functions It needs to be emphasized that many statements made in this section are
motivational and precise proofs of these are well beyond the scope of this text The Fig 5 |
1 | 2143-2146 | In this section, we shall
learn about a new class of (related) functions called exponential functions and logarithmic
functions It needs to be emphasized that many statements made in this section are
motivational and precise proofs of these are well beyond the scope of this text The Fig 5 9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)
= x4 |
1 | 2144-2147 | It needs to be emphasized that many statements made in this section are
motivational and precise proofs of these are well beyond the scope of this text The Fig 5 9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)
= x4 Observe that the curves get steeper as the power of x increases |
1 | 2145-2148 | The Fig 5 9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)
= x4 Observe that the curves get steeper as the power of x increases Steeper the
curve, faster is the rate of growth |
1 | 2146-2149 | 9 gives a sketch of y = f1(x) = x, y = f2(x) = x2, y = f3(x) = x3 and y = f4(x)
= x4 Observe that the curves get steeper as the power of x increases Steeper the
curve, faster is the rate of growth What this means is that for a fixed increment in the
value of x (> 1), the increment in the value of y = fn (x) increases as n increases for n
= 1, 2, 3, 4 |
1 | 2147-2150 | Observe that the curves get steeper as the power of x increases Steeper the
curve, faster is the rate of growth What this means is that for a fixed increment in the
value of x (> 1), the increment in the value of y = fn (x) increases as n increases for n
= 1, 2, 3, 4 It is conceivable that such a statement is true for all positive values of n,
Rationalised 2023-24
MATHEMATICS
126
where fn (x) = xn |
1 | 2148-2151 | Steeper the
curve, faster is the rate of growth What this means is that for a fixed increment in the
value of x (> 1), the increment in the value of y = fn (x) increases as n increases for n
= 1, 2, 3, 4 It is conceivable that such a statement is true for all positive values of n,
Rationalised 2023-24
MATHEMATICS
126
where fn (x) = xn Essentially, this means
that the graph of y = fn (x) leans more
towards the y-axis as n increases |
1 | 2149-2152 | What this means is that for a fixed increment in the
value of x (> 1), the increment in the value of y = fn (x) increases as n increases for n
= 1, 2, 3, 4 It is conceivable that such a statement is true for all positive values of n,
Rationalised 2023-24
MATHEMATICS
126
where fn (x) = xn Essentially, this means
that the graph of y = fn (x) leans more
towards the y-axis as n increases For
example, consider f10(x) = x10 and f15(x)
= x15 |
1 | 2150-2153 | It is conceivable that such a statement is true for all positive values of n,
Rationalised 2023-24
MATHEMATICS
126
where fn (x) = xn Essentially, this means
that the graph of y = fn (x) leans more
towards the y-axis as n increases For
example, consider f10(x) = x10 and f15(x)
= x15 If x increases from 1 to 2, f10
increases from 1 to 210 whereas f15
increases from 1 to 215 |
1 | 2151-2154 | Essentially, this means
that the graph of y = fn (x) leans more
towards the y-axis as n increases For
example, consider f10(x) = x10 and f15(x)
= x15 If x increases from 1 to 2, f10
increases from 1 to 210 whereas f15
increases from 1 to 215 Thus, for the same
increment in x, f15 grow faster than f10 |
1 | 2152-2155 | For
example, consider f10(x) = x10 and f15(x)
= x15 If x increases from 1 to 2, f10
increases from 1 to 210 whereas f15
increases from 1 to 215 Thus, for the same
increment in x, f15 grow faster than f10 Upshot of the above discussion is that
the growth of polynomial functions is
dependent on the degree of the polynomial
function – higher the degree, greater is
the growth |
1 | 2153-2156 | If x increases from 1 to 2, f10
increases from 1 to 210 whereas f15
increases from 1 to 215 Thus, for the same
increment in x, f15 grow faster than f10 Upshot of the above discussion is that
the growth of polynomial functions is
dependent on the degree of the polynomial
function – higher the degree, greater is
the growth The next natural question is:
Is there a function which grows faster than any polynomial function |
1 | 2154-2157 | Thus, for the same
increment in x, f15 grow faster than f10 Upshot of the above discussion is that
the growth of polynomial functions is
dependent on the degree of the polynomial
function – higher the degree, greater is
the growth The next natural question is:
Is there a function which grows faster than any polynomial function The answer is in
affirmative and an example of such a function is
y = f(x) = 10x |
1 | 2155-2158 | Upshot of the above discussion is that
the growth of polynomial functions is
dependent on the degree of the polynomial
function – higher the degree, greater is
the growth The next natural question is:
Is there a function which grows faster than any polynomial function The answer is in
affirmative and an example of such a function is
y = f(x) = 10x Our claim is that this function f grows faster than fn (x) = xn for any positive integer n |
1 | 2156-2159 | The next natural question is:
Is there a function which grows faster than any polynomial function The answer is in
affirmative and an example of such a function is
y = f(x) = 10x Our claim is that this function f grows faster than fn (x) = xn for any positive integer n For example, we can prove that 10x grows faster than f100 (x) = x100 |
1 | 2157-2160 | The answer is in
affirmative and an example of such a function is
y = f(x) = 10x Our claim is that this function f grows faster than fn (x) = xn for any positive integer n For example, we can prove that 10x grows faster than f100 (x) = x100 For large values
of x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) =
10103
= 101000 |
1 | 2158-2161 | Our claim is that this function f grows faster than fn (x) = xn for any positive integer n For example, we can prove that 10x grows faster than f100 (x) = x100 For large values
of x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) =
10103
= 101000 Clearly f (x) is much greater than f100 (x) |
1 | 2159-2162 | For example, we can prove that 10x grows faster than f100 (x) = x100 For large values
of x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) =
10103
= 101000 Clearly f (x) is much greater than f100 (x) It is not difficult to prove that for all
x > 103, f (x) > f100 (x) |
1 | 2160-2163 | For large values
of x like x = 103, note that f100 (x) = (103)100 = 10300 whereas f(103) =
10103
= 101000 Clearly f (x) is much greater than f100 (x) It is not difficult to prove that for all
x > 103, f (x) > f100 (x) But we will not attempt to give a proof of this here |
1 | 2161-2164 | Clearly f (x) is much greater than f100 (x) It is not difficult to prove that for all
x > 103, f (x) > f100 (x) But we will not attempt to give a proof of this here Similarly, by
choosing large values of x, one can verify that f(x) grows faster than fn (x) for any
positive integer n |
1 | 2162-2165 | It is not difficult to prove that for all
x > 103, f (x) > f100 (x) But we will not attempt to give a proof of this here Similarly, by
choosing large values of x, one can verify that f(x) grows faster than fn (x) for any
positive integer n Definition 3 The exponential function with positive base b > 1 is the function
y = f (x) = bx
The graph of y = 10x is given in the Fig 5 |
1 | 2163-2166 | But we will not attempt to give a proof of this here Similarly, by
choosing large values of x, one can verify that f(x) grows faster than fn (x) for any
positive integer n Definition 3 The exponential function with positive base b > 1 is the function
y = f (x) = bx
The graph of y = 10x is given in the Fig 5 9 |
1 | 2164-2167 | Similarly, by
choosing large values of x, one can verify that f(x) grows faster than fn (x) for any
positive integer n Definition 3 The exponential function with positive base b > 1 is the function
y = f (x) = bx
The graph of y = 10x is given in the Fig 5 9 It is advised that the reader plots this graph for particular values of b like 2, 3 and 4 |
1 | 2165-2168 | Definition 3 The exponential function with positive base b > 1 is the function
y = f (x) = bx
The graph of y = 10x is given in the Fig 5 9 It is advised that the reader plots this graph for particular values of b like 2, 3 and 4 Following are some of the salient features of the exponential functions:
(1)
Domain of the exponential function is R, the set of all real numbers |
1 | 2166-2169 | 9 It is advised that the reader plots this graph for particular values of b like 2, 3 and 4 Following are some of the salient features of the exponential functions:
(1)
Domain of the exponential function is R, the set of all real numbers (2)
Range of the exponential function is the set of all positive real numbers |
1 | 2167-2170 | It is advised that the reader plots this graph for particular values of b like 2, 3 and 4 Following are some of the salient features of the exponential functions:
(1)
Domain of the exponential function is R, the set of all real numbers (2)
Range of the exponential function is the set of all positive real numbers (3)
The point (0, 1) is always on the graph of the exponential function (this is a
restatement of the fact that b0 = 1 for any real b > 1) |
1 | 2168-2171 | Following are some of the salient features of the exponential functions:
(1)
Domain of the exponential function is R, the set of all real numbers (2)
Range of the exponential function is the set of all positive real numbers (3)
The point (0, 1) is always on the graph of the exponential function (this is a
restatement of the fact that b0 = 1 for any real b > 1) (4)
Exponential function is ever increasing; i |
1 | 2169-2172 | (2)
Range of the exponential function is the set of all positive real numbers (3)
The point (0, 1) is always on the graph of the exponential function (this is a
restatement of the fact that b0 = 1 for any real b > 1) (4)
Exponential function is ever increasing; i e |
1 | 2170-2173 | (3)
The point (0, 1) is always on the graph of the exponential function (this is a
restatement of the fact that b0 = 1 for any real b > 1) (4)
Exponential function is ever increasing; i e , as we move from left to right, the
graph rises above |
1 | 2171-2174 | (4)
Exponential function is ever increasing; i e , as we move from left to right, the
graph rises above Fig 5 |
1 | 2172-2175 | e , as we move from left to right, the
graph rises above Fig 5 9
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
127
(5)
For very large negative values of x, the exponential function is very close to 0 |
1 | 2173-2176 | , as we move from left to right, the
graph rises above Fig 5 9
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
127
(5)
For very large negative values of x, the exponential function is very close to 0 In
other words, in the second quadrant, the graph approaches x-axis (but never
meets it) |
1 | 2174-2177 | Fig 5 9
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
127
(5)
For very large negative values of x, the exponential function is very close to 0 In
other words, in the second quadrant, the graph approaches x-axis (but never
meets it) Exponential function with base 10 is called the common exponential function |
1 | 2175-2178 | 9
Rationalised 2023-24
CONTINUITY AND DIFFERENTIABILITY
127
(5)
For very large negative values of x, the exponential function is very close to 0 In
other words, in the second quadrant, the graph approaches x-axis (but never
meets it) Exponential function with base 10 is called the common exponential function In
the Appendix A |
1 | 2176-2179 | In
other words, in the second quadrant, the graph approaches x-axis (but never
meets it) Exponential function with base 10 is called the common exponential function In
the Appendix A 1 |
1 | 2177-2180 | Exponential function with base 10 is called the common exponential function In
the Appendix A 1 4 of Class XI, it was observed that the sum of the series
1
1
1 |
1 | 2178-2181 | In
the Appendix A 1 4 of Class XI, it was observed that the sum of the series
1
1
1 1 |
1 | 2179-2182 | 1 4 of Class XI, it was observed that the sum of the series
1
1
1 1 2 |
1 | 2180-2183 | 4 of Class XI, it was observed that the sum of the series
1
1
1 1 2 +
+
+
is a number between 2 and 3 and is denoted by e |
1 | 2181-2184 | 1 2 +
+
+
is a number between 2 and 3 and is denoted by e Using this e as the base we obtain an
extremely important exponential function y = ex |
1 | 2182-2185 | 2 +
+
+
is a number between 2 and 3 and is denoted by e Using this e as the base we obtain an
extremely important exponential function y = ex This is called natural exponential function |
1 | 2183-2186 | +
+
+
is a number between 2 and 3 and is denoted by e Using this e as the base we obtain an
extremely important exponential function y = ex This is called natural exponential function It would be interesting to know if the inverse of the exponential function exists and
has nice interpretation |
1 | 2184-2187 | Using this e as the base we obtain an
extremely important exponential function y = ex This is called natural exponential function It would be interesting to know if the inverse of the exponential function exists and
has nice interpretation This search motivates the following definition |
1 | 2185-2188 | This is called natural exponential function It would be interesting to know if the inverse of the exponential function exists and
has nice interpretation This search motivates the following definition Definition 4 Let b > 1 be a real number |
1 | 2186-2189 | It would be interesting to know if the inverse of the exponential function exists and
has nice interpretation This search motivates the following definition Definition 4 Let b > 1 be a real number Then we say logarithm of a to base b is x if
bx = a |
1 | 2187-2190 | This search motivates the following definition Definition 4 Let b > 1 be a real number Then we say logarithm of a to base b is x if
bx = a Logarithm of a to base b is denoted by logb a |
1 | 2188-2191 | Definition 4 Let b > 1 be a real number Then we say logarithm of a to base b is x if
bx = a Logarithm of a to base b is denoted by logb a Thus logb a = x if bx = a |
1 | 2189-2192 | Then we say logarithm of a to base b is x if
bx = a Logarithm of a to base b is denoted by logb a Thus logb a = x if bx = a Let us
work with a few explicit examples to get a feel for this |
1 | 2190-2193 | Logarithm of a to base b is denoted by logb a Thus logb a = x if bx = a Let us
work with a few explicit examples to get a feel for this We know 23 = 8 |
1 | 2191-2194 | Thus logb a = x if bx = a Let us
work with a few explicit examples to get a feel for this We know 23 = 8 In terms of
logarithms, we may rewrite this as log2 8 = 3 |
1 | 2192-2195 | Let us
work with a few explicit examples to get a feel for this We know 23 = 8 In terms of
logarithms, we may rewrite this as log2 8 = 3 Similarly, 104 = 10000 is equivalent to
saying log10 10000 = 4 |
1 | 2193-2196 | We know 23 = 8 In terms of
logarithms, we may rewrite this as log2 8 = 3 Similarly, 104 = 10000 is equivalent to
saying log10 10000 = 4 Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or
log25 625 = 2 |
1 | 2194-2197 | In terms of
logarithms, we may rewrite this as log2 8 = 3 Similarly, 104 = 10000 is equivalent to
saying log10 10000 = 4 Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or
log25 625 = 2 On a slightly more mature note, fixing a base b > 1, we may look at logarithm as
a function from positive real numbers to all real numbers |
1 | 2195-2198 | Similarly, 104 = 10000 is equivalent to
saying log10 10000 = 4 Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or
log25 625 = 2 On a slightly more mature note, fixing a base b > 1, we may look at logarithm as
a function from positive real numbers to all real numbers This function, called the
logarithmic function, is defined by
logb : R+ → R
x → logb x = y if by = x
As before if the base b = 10, we say it
is common logarithms and if b = e, then
we say it is natural logarithms |
1 | 2196-2199 | Also, 625 = 54 = 252 is equivalent to saying log5 625 = 4 or
log25 625 = 2 On a slightly more mature note, fixing a base b > 1, we may look at logarithm as
a function from positive real numbers to all real numbers This function, called the
logarithmic function, is defined by
logb : R+ → R
x → logb x = y if by = x
As before if the base b = 10, we say it
is common logarithms and if b = e, then
we say it is natural logarithms Often
natural logarithm is denoted by ln |
1 | 2197-2200 | On a slightly more mature note, fixing a base b > 1, we may look at logarithm as
a function from positive real numbers to all real numbers This function, called the
logarithmic function, is defined by
logb : R+ → R
x → logb x = y if by = x
As before if the base b = 10, we say it
is common logarithms and if b = e, then
we say it is natural logarithms Often
natural logarithm is denoted by ln In this
chapter, log x denotes the logarithm
function to base e, i |
1 | 2198-2201 | This function, called the
logarithmic function, is defined by
logb : R+ → R
x → logb x = y if by = x
As before if the base b = 10, we say it
is common logarithms and if b = e, then
we say it is natural logarithms Often
natural logarithm is denoted by ln In this
chapter, log x denotes the logarithm
function to base e, i e |
1 | 2199-2202 | Often
natural logarithm is denoted by ln In this
chapter, log x denotes the logarithm
function to base e, i e , ln x will be written
as simply log x |
1 | 2200-2203 | In this
chapter, log x denotes the logarithm
function to base e, i e , ln x will be written
as simply log x The Fig 5 |
1 | 2201-2204 | e , ln x will be written
as simply log x The Fig 5 10 gives the plots
of logarithm function to base 2, e and 10 |
1 | 2202-2205 | , ln x will be written
as simply log x The Fig 5 10 gives the plots
of logarithm function to base 2, e and 10 Some of the important observations
about the logarithm function to any base
b > 1 are listed below:
Fig 5 |
1 | 2203-2206 | The Fig 5 10 gives the plots
of logarithm function to base 2, e and 10 Some of the important observations
about the logarithm function to any base
b > 1 are listed below:
Fig 5 10
Rationalised 2023-24
MATHEMATICS
128
Fig 5 |
1 | 2204-2207 | 10 gives the plots
of logarithm function to base 2, e and 10 Some of the important observations
about the logarithm function to any base
b > 1 are listed below:
Fig 5 10
Rationalised 2023-24
MATHEMATICS
128
Fig 5 11
(1)
We cannot make a meaningful definition of logarithm of non-positive numbers
and hence the domain of log function is R+ |
1 | 2205-2208 | Some of the important observations
about the logarithm function to any base
b > 1 are listed below:
Fig 5 10
Rationalised 2023-24
MATHEMATICS
128
Fig 5 11
(1)
We cannot make a meaningful definition of logarithm of non-positive numbers
and hence the domain of log function is R+ (2)
The range of log function is the set of all real numbers |
1 | 2206-2209 | 10
Rationalised 2023-24
MATHEMATICS
128
Fig 5 11
(1)
We cannot make a meaningful definition of logarithm of non-positive numbers
and hence the domain of log function is R+ (2)
The range of log function is the set of all real numbers (3)
The point (1, 0) is always on the graph of the log function |
1 | 2207-2210 | 11
(1)
We cannot make a meaningful definition of logarithm of non-positive numbers
and hence the domain of log function is R+ (2)
The range of log function is the set of all real numbers (3)
The point (1, 0) is always on the graph of the log function (4)
The log function is ever increasing,
i |
1 | 2208-2211 | (2)
The range of log function is the set of all real numbers (3)
The point (1, 0) is always on the graph of the log function (4)
The log function is ever increasing,
i e |
1 | 2209-2212 | (3)
The point (1, 0) is always on the graph of the log function (4)
The log function is ever increasing,
i e , as we move from left to right
the graph rises above |
1 | 2210-2213 | (4)
The log function is ever increasing,
i e , as we move from left to right
the graph rises above (5)
For x very near to zero, the value
of log x can be made lesser than
any given real number |
1 | 2211-2214 | e , as we move from left to right
the graph rises above (5)
For x very near to zero, the value
of log x can be made lesser than
any given real number In other
words in the fourth quadrant the
graph approaches y-axis (but
never meets it) |
1 | 2212-2215 | , as we move from left to right
the graph rises above (5)
For x very near to zero, the value
of log x can be made lesser than
any given real number In other
words in the fourth quadrant the
graph approaches y-axis (but
never meets it) (6)
Fig 5 |
1 | 2213-2216 | (5)
For x very near to zero, the value
of log x can be made lesser than
any given real number In other
words in the fourth quadrant the
graph approaches y-axis (but
never meets it) (6)
Fig 5 11 gives the plot of y = ex and
y = ln x |
1 | 2214-2217 | In other
words in the fourth quadrant the
graph approaches y-axis (but
never meets it) (6)
Fig 5 11 gives the plot of y = ex and
y = ln x It is of interest to observe
that the two curves are the mirror
images of each other reflected in the line y = x |
1 | 2215-2218 | (6)
Fig 5 11 gives the plot of y = ex and
y = ln x It is of interest to observe
that the two curves are the mirror
images of each other reflected in the line y = x Two properties of ‘log’ functions are proved below:
(1)
There is a standard change of base rule to obtain loga p in terms of logb p |
1 | 2216-2219 | 11 gives the plot of y = ex and
y = ln x It is of interest to observe
that the two curves are the mirror
images of each other reflected in the line y = x Two properties of ‘log’ functions are proved below:
(1)
There is a standard change of base rule to obtain loga p in terms of logb p Let
loga p = α, logb p = β and logb a = γ |
1 | 2217-2220 | It is of interest to observe
that the two curves are the mirror
images of each other reflected in the line y = x Two properties of ‘log’ functions are proved below:
(1)
There is a standard change of base rule to obtain loga p in terms of logb p Let
loga p = α, logb p = β and logb a = γ This means aα = p, bβ = p and bγ = a |
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