This document discusses properties of exponentials and logarithms. It covers topics such as: - Definitions of exponentials and logarithms - Properties including logb(0), logb(1), logb(b), logb(bx), and logb(y) being undefined for certain values of b and y - Proofs of properties including logb(bx) = x, ylogb(y) = y, log of powers, roots, products, quotients, and changing bases - Over 10 pages of examples are provided to illustrate each property

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

3.
Enzo Exposyto 3

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

6. LOGARITHMS
-
THEIR PROPERTIES
Enzo Exposyto 34

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