#6 - LOGARITHMS LOG of a POWER LOG of a ROOT PROOFS 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 MATHS SYMBOLS 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. logb(bx) = x
PROOFS
Enzo Exposyto 61

7. logb(bx) = x
a) The log of bx is the exponent which we have to put on the
base b to get bx itself and, therefore, it’s “x”
b) If,
from x, we ‘go’ to bx
and, then,
from bx, we ‘go’ to logb(bx),
since
logb(bx) is the inverse operation of bx,
we go back to x …
therefore,
logb(bx) = x
In other words
(remembering that bx = antilogb(x)):
logb(antilogb(x)) = x
Enzo Exposyto 62

8. logb(bx) = x
c) Let's set
bx = y
and, then, we ‘do’ the logarithms in base b of both sides;
we get
logb(bx) = logb(y)
Since
bx = y <=> logb(y) = x
we can write
logb(bx) = logb(y)
= x
Therefore,
logb(bx) = x
Q.E.D.
Enzo Exposyto 63

9. blogb(y) = y
PROOFS
Enzo Exposyto 64

10. blogb(y) = y
a) If
from y, we ‘go’ to logb(y)
and, then,
from logb(y), we ‘go’ to blogb(y),
since
blogb(y) is the inverse operation of logb(y),
we go back to y …
therefore,
blogb(y) = y
In other words
(remembering that blogb(y) = antilogb(logb(y))):
antilogb(logb(y)) = y
Enzo Exposyto 65

11. blogb(y) = y
b) Let's set
bx = y
and, then, we ‘do’ the log in base b of both sides; we get
logb(bx) = logb(y)
Remembering that (pages 62-63)
logb(bx) = x
we can write
logb(bx) = logb(y)
x = logb(y)
or
logb(y) = x
Now, we ‘do’ the exponentials in base b of both sides and we get
blogb(y) = bx
Since
bx = y
we get
blogb(y) = y
Q.E.D.
Enzo Exposyto 66

12. blogb(y) = y
c) Let's set
logb(y) = x
and, then,
on the left hand side of the equation,
we get
blogb(y) = bx
Since
logb(y) = x <=> bx = y
it's
blogb(y) = bx = y
and, therefore,
blogb(y) = y
Q.E.D.
Enzo Exposyto 67

13. log of
a POWER
PROOFS
Enzo Exposyto 68

14. logb(yz) = z . logb(y)
1) Let's set
logb(y) = l
and, then, the right hand side of the equation becomes
z . logb(y) = z . l
2) Since
logb(y) = l <=> bl = y
or y = bl
the left hand side of the equation becomes
logb(yz) = logb((bl)z)
= logb(blz)
Remembering that (pages 62-63)
logb(bx) = x
it's
logb(blz) = lz = z . l
and we can write
logb(yz) = logb(blz) = z . l
3) Since the left hand side and the right hand side are equal to z . l, they are equal:
logb(yz) = z . logb(y)
Q.E.D.
Enzo Exposyto 69

15. logb(1) = - logb(y)
y
OR
- logb(y) = logb(1)
y
Remembering that
logb(yz) = z . logb(y) [log of a Power, previous page]
we get
logb(1) = logb(y-1)
y
= (-1) . logb(y)
= - logb(y)
Q.E.D.
Enzo Exposyto 70

16. log of
a ROOT
PROOFS
Enzo Exposyto 71

17. logb(n√y) = logb(y)
n
Remembering that
logb(yz) = z . logb(y) [log of a Power, page 69]
we get
logb(n√y) = logb(y1/n)
= 1 logb(y)
n
= logb(y)
n
Q.E.D.
Enzo Exposyto 72

18. logb(n√yz) = z . logb(y)
n
Remembering that
logb(yz) = z . logb(y) [log of a Power, page 69]
we get
logb(n√yz) = logb(yz/n)
= z logb(y)
n
= z . logb(y)
n
Q.E.D.
Enzo Exposyto 73