MATHS SYMBOLS #1 - EXPONENTIALS and THEIR PROPERTIES DEFINITION with 6+ PROPERTIES ANTILOGARITHMS INVERSE OPERATIONS NATURAL LOGARITHMS NEPER - NAPIER - EULER'S NUMBER LOG - LN POWER - ROOT PRODUCT - QUOTIENT CHANGE of BASE PROOFS - EXAMPLES CALCULATIONS STEP by STEP www.enzoexposito.it/mobile/matematica.html www.enzoexposito.it/mobile/expon_log.html

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

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. EXPONENTIAL
-
DEFINITION
Enzo Exposyto 6

7. EXPONENTIAL - DEFINITION
DEFINITION Examples
DEFINITION
bx = y ...
2-1 = 1
2
...
20 = 1
...
21 = 2
...
b > 0 and b < > 1 (*)
x element of R
y > 0
"b" is the fixed base
x is the variable exponent
(*) The symbol < > and ≠ have the same meaning
Enzo Exposyto 7

8. EXPONENTIALS
-
THEIR PROPERTIES
Enzo Exposyto 8

9. EXPONENTIALS - THEIR PROPERTIES - 1
[b, c > 0 and b, c < > 1]
[x, y elements of R]
Z+ = {1, 2, 3, …}
Property Exponentials Exponents Exponentials Exponents Result
1st bx · by = bx + y 23 · 22 = 23 + 2 = 32
2nd
bx
=
by
bx -y
23
=
22
23 -2 = 2
3rd (bx)y = bx *y (23)2 = 23 *2 = 64
4th
n Z+ n√bx = bx : n 2√24 = 24 : 2 = 4
5th bx · cx = (b · c)x 22 · 32 = (2 · 3)2 = 36
6th
bx
———- =
cx
(_b_)x
c
43
———- =
23
(_4_)3
=
2
8
Enzo Exposyto 9

10. EXPONENTIALS - THEIR PROPERTIES - 2
[b, c > 0 and b, c < > 1]
[x, y elements of R]
Z+ = {1, 2, 3, …}
Property Exponents Exponentials Exponents Exponentials Result
1st bx + y = bx · by
23 + 2 = 23 · 22 = 32
2nd bx -y =
bx
———-
by
23 -2 =
23
=
22
2
3rd bx *y = (bx)y 23 *2 = (23)2 = 64
4th n Z+ bx : n = n√bx
24 : 2 = 2√24 = 4
5th (b · c)x = bx · cx (2 · 3)2 = 22 · 32 = 36
6th
(_b_)x =
c
bx
———-
cx
(_4_)3
=
2
43
———- =
23
8
Enzo Exposyto 10

11. Exponentials - 6 Properties - Proofs/Examples
PROOFS / EXAMPLES
[b, c > 0 and b, c < > 1]
1st b3 · b2 = (b · b · b) · (b · b) = b · b · b · b · b = b5 = b3 + 2
2nd b3 = (b · b · b) = b = b1 = b3 - 2
b2 (b · b)
3rd (b3)2 = (b · b · b) · (b · b · b) = b · b · b · b · b · b = b6 = b3 * 2
4th 2√b4 = 2√(b · b · b · b) = b · b = b2 = b4 : 2
5th b3 · c3 = (b · b · b) · (c · c · c) = b · c · b · c · b · c = … = (b·c)3
6th
b3 (b · b · b) b b b b
——- = ————— = —— . —— . —— = … = (—)3
c3 (c · c · c) c c c c
Enzo Exposyto 11

12. Exponentials - 2nd property - b superscript 0
EXPONENTIALS - THEIR PROPERTIES - 3
Reference Property Notice Proof Example
2nd
Property
b0 = 1
b > 0
b ≠ 1
x R
b0 = bx - x 20 = 23 - 3
= bx
bx
= 23
23
= 1 = 1
1 = b0 b > 0
b ≠ 1
1 = 20
Enzo Exposyto 12

13. Exponentials - 2nd property - b superscript (-x)
EXPONENTIALS - THEIR PROPERTIES - 4
Reference Property Notice Proof Example
2nd
Property
b-x = 1
bx
b > 0
b ≠ 1
x R
b-x = b0-x 2-3 = 20-3
= b0
bx
= 20
23
= 1
bx
= 1
23
1 = b-x
bx
b > 0
b ≠ 1
x R
1 = 2-3
23
Enzo Exposyto 13

14. Exponentials - 2nd property - Reciprocal of bx
EXPONENTIALS and THEIR PROPERTIES - 5
Reference Property Notice Proof Example
2nd
Property
bx = 1
b-x
b > 0
b ≠ 1
x R
23 = 1
2-3
1 = bx
b-x
b > 0
b ≠ 1
x R
1 = 23
2-3
Enzo Exposyto 14