MATHS SYMBOLS #2 - EXPONENTIALS and LOGARITHMS DEFINITION of LOGARITHM 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. EXPONENTIALS
and
LOGARITHMS
Enzo Exposyto 15

7. EXPONENTIAL BASE 2:
23 = 2 × 2 x 2 = 8
The LOGARITHM BASE 2 OF 8
goes the OTHER WAY:
Enzo Exposyto 16

8. The logarithm base 2 of 8 is 3,
BECAUSE
2 cubed is 8;
so the logarithm base 2 of 8 is 3:
Enzo Exposyto 17

9. THE LOGARITHM BASE 2 OF 8
IS THE OPERATION THAT ALLOWS US
OF GOING BACK TO THE EXPONENT 3
EXPONENTIAL 2x and LOGARITHM BASE 2
EXPONENT x 2x
1 2
2 4
3 8
4 16
Enzo Exposyto 18

10. In other words …
THE LOGARITHM BASE 2 OF 8
IS
THE EXPONENT 3
… MORE PRECISELY …
THE LOGARITHM "3"
IS THE EXPONENT
WHICH WE HAVE TO PUT ON
THE BASE “2”
TO GET “8”
Enzo Exposyto 19

11. Now, since
log2(8) = 3 and 8 = 23
then
log2(23) = 3
We can see that
THE LOGARITHM "3"
IS THE EXPONENT
WHICH WE HAVE TO PUT ON
THE BASE “2”
TO GET “23”
Enzo Exposyto 20

12. EXPONENTIAL and LOGARITHM
with the same base
CANCEL EACH OTHER.
This is true because
exponential and logarithm
with the same base
are INVERSE OPERATIONS
It is just like
Addition and Subtraction,
Multiplication and Division,
Exponentiation and Root, …
when they're
INVERSE OPERATIONS
Enzo Exposyto 21

13. Now, we can introduce
the ANTILOGARITHM BASE b:
antilogb(x) = bx
It's, simply, an EXPONENTIAL
and represents the antilogarithm
when we operate
with a logarithm …
It’s such that
logb(antilogb(x)) = x
The meaning is
logb(bx) = x
Enzo Exposyto 22

14. And, of course,
antilogb(logb(y)) = y
The meaning is
blogb(y) = y
Enzo Exposyto 23

15. These phrases - with 8 - are equivalent
log2(8) = 3 < = > 23 = 8
OR
23 = 8 < = > log2(8) = 3
These phrases - with 23 - are equivalent
log2(23) = 3 < = > 23 = 23
OR
23 = 23 < = > log2(23) = 3
Enzo Exposyto 24