MATHS SYMBOLS #3 - LOGARITHMS DEFINITION of LOGARITHM EXAMPLES 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. LOGARITHM
-
DEFINITION
and
EXAMPLES
Enzo Exposyto 25

7. LOGARITHM - DEFINITION
DEFINITION Example
DEFINITION
logb(y) = x
iff (if and only if)
bx = y
log2(8) = 3
because
23 = 8
b > 0 and b < > 1
y > 0
x element of R
2 > 0 and 2 < > 1
8 > 0
3 element of R
The logarithm "x" is the exponent
which we have to put on
the base “b”
to get “y”
The logarithm
"3"
is the exponent
which we have
to put on
the base “2”
to get “8”
Enzo Exposyto 26

8. iff
bx = y Enzo Exposyto 27

9. LOGARITHMS - EXAMPLES - 1
log DEFINITION because
log3(9) = 2
The logarithm "2" is the exponent
which we have to put on the base “3”
to get “9”
log3(9) = 2
because
32 = 9
log3(27) = 3
The logarithm "3" is the exponent
which we have to put on the base “3”
to get “27”
log3(27) = 3
because
33 = 27
log4(4) = 1
The logarithm "1" is the exponent
which we have to put on the base “4”
to get “(4)”
log4(4) = 1
because
41 = (4)
Enzo Exposyto 28

10. LOGARITHMS - EXAMPLES - 2
log DEFINITION because
log4(16) = 2
The logarithm "2" is the exponent
which we have to put on the base “4”
to get “16”
log4(16) = 2
because
42 = 16
log5(25) = 2
The logarithm "2" is the exponent
which we have to put on the base “5”
to get “25”
log5(25) = 2
because
52 = 25
log10(100) = 2
The logarithm "2" is the exponent
which we have to put on the base “10”
to get “100”
log10(100) = 2
because
102 = 100
Enzo Exposyto 29

11. LOGARITHMS - EXAMPLES - 3
log DEFINITION because
log4(1) = -1
4
The logarithm "-1" is the exponent
which we have to put on the base “4”
to get “1”
4
log4(1) = -1
4
because
4-1 = 1
4
log10( 1 ) = -1
10
The logarithm "-1" is the exponent
which we have to put on the base “10”
to get “ 1 ”
10
log10( 1 ) = -1
10
because
10-1 = 1
10
Enzo Exposyto 30

12. LOGARITHMS - EXAMPLES - 4
log DEFINITION because
log1/2(2) = -1
The logarithm "-1" is the exponent
which we have to put on the base “1”
2
to get “2”
log1/2(2) = -1
because
(1)-1 = 2
2
log1/4(4) = -1
The logarithm "-1" is the exponent
which we have to put on the base “1”
4
to get “4”
log1/4(4) = -1
because
(1)-1 = 4
4
Enzo Exposyto 31

13. LOGARITHMS - EXAMPLES - 5
log DEFINITION because
log1/2(4) = -2
The logarithm "-2" is the exponent
which we have to put on the base “1”
2
to get “4”
log1/2(4) = -2
because
(1)-2 = 4
2
log1/3(9) = -2
The logarithm "-2" is the exponent
which we have to put on the base “1”
3
to get “9”
log1/3(9) = -2
because
(1)-2 = 9
3
Enzo Exposyto 32

14. LOGARITHMS - EXAMPLES - 6
e = 2.718281828…
log DEFINITION because
loge(e) = 1
The logarithm "1" is the exponent
which we have to put on the base “e”
to get “(e)”
loge(e) = 1
because
e1 = (e)
loge(1) = 0
The logarithm "0" is the exponent
which we have to put on the base “e”
to get “1”
loge(1) = 0
because
e0 = 1
loge(e2) = 2
The logarithm "2" is the exponent
which we have to put on the base “e”
to get “(e2)”
loge(e2) = 2
because
e2 = (e2)
Enzo Exposyto 33