MATHS SYMBOLS #5 - LOGARITHMS LOG(y) - LN(y) THEIR 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. log(y) AND ln(y)
-
THEIR
PROPERTIES
Enzo Exposyto 46

7. log(y) AND ln(y) - THEIR PROPERTIES
REMARKS:
• log(y) always refers to log base 10,
i. e.,
log(y) = log10(y)
Therefore,
log(y) = x
if and only if
10x = y
Enzo Exposyto 47

8. • ln(y) is called the natural logarithm
and is used to represent loge(y),
where the irrational number e 2.718281828:
ln(y) = loge(y)
Therefore,
ln(y) = x
if and only if
ex = y
Enzo Exposyto 48

9. • Most calculators can directly compute
logs base 10
and/or
the natural log.
For any other base
it is necessary to use
the change of the base formula:
logb(y) = log10(y) = log(y) log2(8) = log(8)
log10(b) log(b). log(2)
or
logb(y) = ln(y) log2(8) = ln(8)
ln(b) ln(2)
Enzo Exposyto 49

10. log(y) AND ln(y) - THEIR PROPERTIES - 1
Property log ln
base = 0
base = 10 base = e
base = 1
base = 10 base = e
logb(0)
log(0) is undefined ln(0) is undefined
For any x R, 10x ≠ 0
(really 10x > 0) and, then,
log(0) Does Not Exist
For any x R, ex ≠ 0
(really ex > 0) and, then,
ln(0) Does Not Exist
Enzo Exposyto 50

11. log(x) AND ln(x) - THEIR PROPERTIES - 2
Property log ln
logb(1)
log(1) = 0 ln(1) = 0
because
100 = 1
because
e0 = 1
logb(b)
log(10) = 1 ln(e) = 1
because
101 = 10
because
e1 = e
Enzo Exposyto 51

12. log(y) AND ln(y) - THEIR PROPERTIES - 3
Property log ln
logb(bx)
log(10x) = x ln(ex) = x
x element of R x R
blogb(y)
10log(y) = y eln(y) = y
y > 0 y > 0
Enzo Exposyto 52

13. log(y) AND ln(y) - THEIR PROPERTIES - 4
Property log ln
log of a
Power
log(yz) = z log(y) ln(yz) = z ln(y)
y > 0
z element of R
y > 0
z element of R
log of
a Reciprocal
log(1) = log(y-1) = - log(y)
y
ln(1) = ln(y-1) = - ln(y)
y
y > 0 y > 0
Enzo Exposyto 53

14. log(y) AND ln(y) - THEIR PROPERTIES - 5
Property log ln
log of
a Root - 1
log(n√y) = log(y)
n
ln(n√y) = ln(y)
n
n Z+
Z+ = {1, 2, 3, …}
y > 0
n Z+
Z+ = {1, 2, 3, …}
y > 0
log of
a Root - 2
log(n√yz) = z log(y)
n
ln(n√yz) = z ln(y)
n
n element of Z+
Z+ = {1, 2, 3, …}
y > 0
z element of R
n element of Z+
Z+ = {1, 2, 3, …}
y > 0
z element of R
Enzo Exposyto 54

15. log(y) AND ln(y) - THEIR PROPERTIES - 6
Property log ln
log of a
Product - 1
log(y z) = log(y) + log(z) ln(y z) = ln(y) + ln(z)
y, z > 0 y, z > 0
log of a
Product - 2
log(yn . zp) = n.log(y)+p.log(z) ln(yn . zp) = n.ln(y)+p.ln(z)
y, z > 0
n, p elements of R
y, z > 0
n, p elements of R
Enzo Exposyto 55

16. log(y) AND ln(y) - THEIR PROPERTIES - 7
Property log ln
log of a
Quotient - 1
log(y) = log(y.z-1) = log(y)-log(z)
z
ln(y) = ln(y.z-1) = ln(y)-ln(z)
z
y, z > 0 y, z > 0
log of a
Quotient - 2
log(y) = log(y) - log(z)
z
ln(y) = ln(y) - ln(z)
z
y, z > 0 y, z > 0
log of a
Quotient - 3
log(yn) = n.log(y)-p.log(z)
zp
ln(yn) = n.ln(y)-p.ln(z)
zp
y, z > 0
n, p elements of R
y, z > 0
n, p elements of R
Enzo Exposyto 56

17. log(y) AND ln(y) - THEIR PROPERTIES - 8
Property log ln
Base Change - 1
log(y) = logc(y)
logc(10)
ln(y) = logc(y)
logc(e)
y > 0
c > 0 and c < > 1
y > 0
c > 0 and c < > 1
Base Switch - 1
log(c) = logc(c) = 1
logc(10) logc(10)
ln(c) = logc(c) = 1
logc(e) logc(e)
logc(c) = 1 logc(c) = 1
c > 0 and c < > 1 c > 0 and c < > 1
Base Switch - 2
log(c) * logc(10) = 1 ln(c) * logc(e) = 1
c > 0 and c < > 1 c > 0 and c < > 1
Enzo Exposyto 57

18. log10(y) AND loge(y) - THEIR PROPERTIES - 9
Property log10 loge
Base Change - 2
log10n(y) = log10(y)
n
logen(y) = loge(y)
n
n element of R, n < > 0
y > 0
n element of R, n < > 0
y > 0
Base Change - 3a
n . log10n(y) = log10(y) n . logen(y) = loge(y)
n element of R, n < > 0
y > 0
n element of R, n < > 0
y > 0
Base Change - 3b
log10(y) = n . log10n(y) loge(y) = n . logen(y)
y > 0
n element of R, n < > 0
y > 0
n element of R, n < > 0
Enzo Exposyto 58

19. log10(y) AND loge(y) - THEIR PROPERTIES - 10
Property log10 loge
Base Change - 4
log1/10(y) = - log10(y) log1/e(y) = - loge(y)
y > 0 y > 0
Base Change - 5
log10(1) = log1/10(y)
y
loge(1) = log1/e(y)
y
y > 0 y > 0
Enzo Exposyto 59

20. log10(y) AND loge(y) - THEIR PROPERTIES - 11
Property log10 loge
zlogb(y)
zlog10(y) = ylog10(z) zloge(y) = yloge(z)
z > 0
y > 0
z > 0
y > 0
Enzo Exposyto 60