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ENZO EXPOSYTO
MATHS
SYMBOLS
PROPERTIES of EXPONENTIALS and LOGARITHMS

Enzo Exposyto 1
2X ex 2-x e-x
EXPONENTIALS
LOGARITHMS
log2(x) ln(x) log(x)

Enzo Exposyto 2


Enzo Exposyto 3
1 - Exponential - Definition 6
2 - Exponentials - Their Properties 8
3 - Exponentials and Logarithms 15
4 - Logarithm - Definition and Examples 25
5 - Logarithms - Their Properties 34
6 - log(y) and ln(y) - Properties 46
Enzo Exposyto 4
1 - Exponential - Definition 6
2 - Exponentials - Their Properties 8
3 - Exponentials and Logarithms 15
4 - Logarithm - Definition and Examples 25
5 - Logarithms - Their Properties 34
6 - log(y) and ln(y) - Properties 46
Enzo Exposyto 4
7 - logb(bx) = x - Proofs 61
8 - ylogb(y) = y - Proofs 64
9 - log of a Power - Proofs 68
10 - log of a Root - Proofs 71
11 - log of a Product - Proofs 74
12 - log of a Quotient - Proofs 77
13 - Change of Base - Proofs 82
14 - zlogb(y) = ylogb(z) - Proof 91
15 - SitoGraphy 93
Enzo Exposyto 5
LOGARITHMS
-
THEIR PROPERTIES

Enzo Exposyto 34
LOGARITHMS - THEIR PROPERTIES - 1
Name Property Example
base = 0
log0(y) is undefined log0(2) is undefined
y > 0
For any x R-{0}, 0x ≠ 2
(really 0x = 0) and, then,
log0(2) Does Not Exist
base = 1
log1(y) is undefined log1(2) is undefined
y > 0
For any x R, 1x ≠ 2
(really 1x = 1) and, then,
log1(2) Does Not Exist
logb(0)
logb(0) is undefined log2(0) is undefined
b > 0 and b < > 1
For any x R, 2x ≠ 0
(really 2x > 0) and, then,
log2(0) Does Not Exist
Enzo Exposyto 35
LOGARITHMS - THEIR PROPERTIES - 2
Name Property Examples
logb(1)
logb(1) = 0 log2(1) = 0
because
20 = 1b > 0 and b < > 1
logb(b)
logb(b) = 1 log4(4) = 1
because
41 = (4)b > 0 and b < > 1
Enzo Exposyto 36
LOGARITHMS - THEIR PROPERTIES - 3
Name Property Example
logb(bx)
logb(bx) = x log2(23) = 3
because
23 = (23)
b > 0 and b < > 1
x element of R
blogb(y)
blogb(y) = y 2log2(8) = 8
because
23 = 8
b > 0 and b < > 1
y > 0
Enzo Exposyto 37
LOGARITHMS - THEIR PROPERTIES - 4
Name Property Examples
log of
a Power
logb(yz) = z logb(y) log2(42) = 2 log2(4)
b > 0 and b < > 1
y > 0
z element of R
b = 2
y = 4
z = 2
log of
a Reciprocal
logb(1) = logb(y-1) = - logb(y)
y
log2(1) = log2(4-1) = - log2(4)
4
b > 0 and b < > 1
y > 0
b = 2
y = 4
Enzo Exposyto 38
LOGARITHMS - THEIR PROPERTIES - 5
Name Property Examples
log of
a Root - 1
logb(n√y) = logb(y)
n
log2(3√8) = log2(8)
3
b > 0 and b < > 1
y > 0
n Z+
Z+ = {1, 2, 3, …}
b = 2
y = 8
n = 3
log of
a Root - 2
logb(n√yz) = z logb(y)
n
log2(3√26) = 6 log2(2)
3
b > 0 and b < > 1
y > 0
z element of R
n Z+
Z+ = {1, 2, 3, …}
b = 2
y = 2
z = 6
n = 3
Enzo Exposyto 39
LOGARITHMS - THEIR PROPERTIES - 6
Name Property Examples
log of a
Product - 1
logb(y z) = logb(y) + logb(z) log2(4 2) = log2(4) + log2(2)
b > 0 and b < > 1
y, z > 0
b = 2
y = 4; z = 2
log of a
Product - 2
logb(yn . zp) = n.logb(y)+p.logb(z) log2(43 . 24) = 3.log2(4)+4.log2(2)
b > 0 and b < > 1
y, z > 0
n, p elements of R
b = 2
y = 4; z = 2
n = 3; p = 4
Enzo Exposyto 40
LOGARITHMS - THEIR PROPERTIES - 7
Name Property Examples
log of a
Quotient - 1
logb(y) = logb(y.z-1) = logb(y)-logb(z)
z
log2(8) = log2(8.4-1) = log2(8) - log2(4)
4
b > 0 and b < > 1
y, z > 0
b = 2
y = 8; z = 4
log of a
Quotient - 2
logb(y) = logb(y) - logb(z)
z
log2(4) = log2(4) - log2(2)
2
b > 0 and b < > 1
y, z > 0
b = 2
y = 4; z = 2
log of a
Quotient - 3
logb(yn) = n.logb(y)-p.logb(z)
zp
log2(43) = 3.log2(4)-4.log2(2)
24
b > 0 and b < > 1
y, z > 0
n, p elements of R
b = 2
y = 4; z = 2
n = 3; p = 4
Enzo Exposyto 41
LOGARITHMS - THEIR PROPERTIES - 8
Name Property Examples
Base Change - 1
logb(y) = logc(y)
logc(b) log2(16) = log4(16) = 2 = 4
log4(2) 1
2b, c > 0 and b, c < > 1
y > 0
Base Switch - 1
logb(c) = logc(c) = 1
logc(b) logc(b)
log2(4) = log4(4) = 1
log4(2) log4(2)
logc(c) = 1 log4(4) = 1
b, c > 0 and b, c < > 1 b = 2; c= 4
Base Switch - 2 logb(c) * logc(b) = 1 log2(4) * log4(2) = 1
Enzo Exposyto 42
LOGARITHMS - THEIR PROPERTIES - 9
Name Property Examples
Base Change - 2
logbn(y) = logb(y)
n
log2-3(8) = log2(8)
-3
b > 0 and b < > 1
n element of R, n < > 0
y > 0
b = 2
n = -3
y = 8
Base Change - 3a
n . logbn(y) = logb(y) -3 . log2-3(8) = log2(8)
n element of R, n < > 0
b > 0 and b < > 1
y > 0
n = -3
b = 2
y = 8
Base Change - 3b
logb(y) = n . logbn(y) log2(4) = 2 . log22(4)
b > 0 and b < > 1
y > 0
n element of R, n < > 0
b = 2
y = 4
n = 2
Enzo Exposyto 43
LOGARITHMS - THEIR PROPERTIES - 10
Name Property Examples
Base Change - 4
log1/b(y) = - logb(y) log1/8(8) = - log8(8)
b > 0 and b < > 1
y > 0
b = 8
y = 8
Base Change - 5
logb(1) = log1/b(y)
y
log2(1) = log1/2(4)
4
b > 0 and b < > 1
y > 0
b = 2
y = 4
Enzo Exposyto 44
LOGARITHMS - THEIR PROPERTIES - 11
Name Property Examples
zlogb(y)
zlogb(y) = ylogb(z) 2log2(8) = 8log2(2)
z > 0
b > 0 and b < > 1
y > 0
z = 2
b = 2
y = 8
Enzo Exposyto 45

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MATHS SYMBOLS - #4 - LOGARITHMS - THEIR PROPERTIES

  • 1. ENZO EXPOSYTO MATHS SYMBOLS PROPERTIES of EXPONENTIALS and LOGARITHMS
 Enzo Exposyto 1
  • 2. 2X ex 2-x e-x EXPONENTIALS LOGARITHMS log2(x) ln(x) log(x)
 Enzo Exposyto 2
  • 4. 1 - Exponential - Definition 6 2 - Exponentials - Their Properties 8 3 - Exponentials and Logarithms 15 4 - Logarithm - Definition and Examples 25 5 - Logarithms - Their Properties 34 6 - log(y) and ln(y) - Properties 46 Enzo Exposyto 4 1 - Exponential - Definition 6 2 - Exponentials - Their Properties 8 3 - Exponentials and Logarithms 15 4 - Logarithm - Definition and Examples 25 5 - Logarithms - Their Properties 34 6 - log(y) and ln(y) - Properties 46 Enzo Exposyto 4
  • 5. 7 - logb(bx) = x - Proofs 61 8 - ylogb(y) = y - Proofs 64 9 - log of a Power - Proofs 68 10 - log of a Root - Proofs 71 11 - log of a Product - Proofs 74 12 - log of a Quotient - Proofs 77 13 - Change of Base - Proofs 82 14 - zlogb(y) = ylogb(z) - Proof 91 15 - SitoGraphy 93 Enzo Exposyto 5
  • 7. LOGARITHMS - THEIR PROPERTIES - 1 Name Property Example base = 0 log0(y) is undefined log0(2) is undefined y > 0 For any x R-{0}, 0x ≠ 2 (really 0x = 0) and, then, log0(2) Does Not Exist base = 1 log1(y) is undefined log1(2) is undefined y > 0 For any x R, 1x ≠ 2 (really 1x = 1) and, then, log1(2) Does Not Exist logb(0) logb(0) is undefined log2(0) is undefined b > 0 and b < > 1 For any x R, 2x ≠ 0 (really 2x > 0) and, then, log2(0) Does Not Exist Enzo Exposyto 35
  • 8. LOGARITHMS - THEIR PROPERTIES - 2 Name Property Examples logb(1) logb(1) = 0 log2(1) = 0 because 20 = 1b > 0 and b < > 1 logb(b) logb(b) = 1 log4(4) = 1 because 41 = (4)b > 0 and b < > 1 Enzo Exposyto 36
  • 9. LOGARITHMS - THEIR PROPERTIES - 3 Name Property Example logb(bx) logb(bx) = x log2(23) = 3 because 23 = (23) b > 0 and b < > 1 x element of R blogb(y) blogb(y) = y 2log2(8) = 8 because 23 = 8 b > 0 and b < > 1 y > 0 Enzo Exposyto 37
  • 10. LOGARITHMS - THEIR PROPERTIES - 4 Name Property Examples log of a Power logb(yz) = z logb(y) log2(42) = 2 log2(4) b > 0 and b < > 1 y > 0 z element of R b = 2 y = 4 z = 2 log of a Reciprocal logb(1) = logb(y-1) = - logb(y) y log2(1) = log2(4-1) = - log2(4) 4 b > 0 and b < > 1 y > 0 b = 2 y = 4 Enzo Exposyto 38
  • 11. LOGARITHMS - THEIR PROPERTIES - 5 Name Property Examples log of a Root - 1 logb(n√y) = logb(y) n log2(3√8) = log2(8) 3 b > 0 and b < > 1 y > 0 n Z+ Z+ = {1, 2, 3, …} b = 2 y = 8 n = 3 log of a Root - 2 logb(n√yz) = z logb(y) n log2(3√26) = 6 log2(2) 3 b > 0 and b < > 1 y > 0 z element of R n Z+ Z+ = {1, 2, 3, …} b = 2 y = 2 z = 6 n = 3 Enzo Exposyto 39
  • 12. LOGARITHMS - THEIR PROPERTIES - 6 Name Property Examples log of a Product - 1 logb(y z) = logb(y) + logb(z) log2(4 2) = log2(4) + log2(2) b > 0 and b < > 1 y, z > 0 b = 2 y = 4; z = 2 log of a Product - 2 logb(yn . zp) = n.logb(y)+p.logb(z) log2(43 . 24) = 3.log2(4)+4.log2(2) b > 0 and b < > 1 y, z > 0 n, p elements of R b = 2 y = 4; z = 2 n = 3; p = 4 Enzo Exposyto 40
  • 13. LOGARITHMS - THEIR PROPERTIES - 7 Name Property Examples log of a Quotient - 1 logb(y) = logb(y.z-1) = logb(y)-logb(z) z log2(8) = log2(8.4-1) = log2(8) - log2(4) 4 b > 0 and b < > 1 y, z > 0 b = 2 y = 8; z = 4 log of a Quotient - 2 logb(y) = logb(y) - logb(z) z log2(4) = log2(4) - log2(2) 2 b > 0 and b < > 1 y, z > 0 b = 2 y = 4; z = 2 log of a Quotient - 3 logb(yn) = n.logb(y)-p.logb(z) zp log2(43) = 3.log2(4)-4.log2(2) 24 b > 0 and b < > 1 y, z > 0 n, p elements of R b = 2 y = 4; z = 2 n = 3; p = 4 Enzo Exposyto 41
  • 14. LOGARITHMS - THEIR PROPERTIES - 8 Name Property Examples Base Change - 1 logb(y) = logc(y) logc(b) log2(16) = log4(16) = 2 = 4 log4(2) 1 2b, c > 0 and b, c < > 1 y > 0 Base Switch - 1 logb(c) = logc(c) = 1 logc(b) logc(b) log2(4) = log4(4) = 1 log4(2) log4(2) logc(c) = 1 log4(4) = 1 b, c > 0 and b, c < > 1 b = 2; c= 4 Base Switch - 2 logb(c) * logc(b) = 1 log2(4) * log4(2) = 1 Enzo Exposyto 42
  • 15. LOGARITHMS - THEIR PROPERTIES - 9 Name Property Examples Base Change - 2 logbn(y) = logb(y) n log2-3(8) = log2(8) -3 b > 0 and b < > 1 n element of R, n < > 0 y > 0 b = 2 n = -3 y = 8 Base Change - 3a n . logbn(y) = logb(y) -3 . log2-3(8) = log2(8) n element of R, n < > 0 b > 0 and b < > 1 y > 0 n = -3 b = 2 y = 8 Base Change - 3b logb(y) = n . logbn(y) log2(4) = 2 . log22(4) b > 0 and b < > 1 y > 0 n element of R, n < > 0 b = 2 y = 4 n = 2 Enzo Exposyto 43
  • 16. LOGARITHMS - THEIR PROPERTIES - 10 Name Property Examples Base Change - 4 log1/b(y) = - logb(y) log1/8(8) = - log8(8) b > 0 and b < > 1 y > 0 b = 8 y = 8 Base Change - 5 logb(1) = log1/b(y) y log2(1) = log1/2(4) 4 b > 0 and b < > 1 y > 0 b = 2 y = 4 Enzo Exposyto 44
  • 17. LOGARITHMS - THEIR PROPERTIES - 11 Name Property Examples zlogb(y) zlogb(y) = ylogb(z) 2log2(8) = 8log2(2) z > 0 b > 0 and b < > 1 y > 0 z = 2 b = 2 y = 8 Enzo Exposyto 45