Log 0 001

Logarithm Rules

The
base of operations
b
logarithm
of a number is the
exponent
that nosotros demand to raise the
base
in lodge to get the number.

• Logarithm definition
• Logarithm rules
• Logarithm issues
• Complex logarithm
• Graph of log(ten)
• Logarithm table
• Logarithm computer

Logarithm definition

When b is raised to the power of y is equal 10:

b
^y

=
x

And then the base b logarithm of x is equal to y:

log
_b
(x)
= y

For instance when:

2^iv
= sixteen

And so

log₂(16) = 4

Logarithm every bit changed part of exponential function

The logarithmic role,

y
= log
_b
(x)

is the changed function of the exponential office,

x
=
b^y

So if we summate the exponential function of the logarithm of ten (x>0),

f
(f

^-1(x)) =
b
^log
b
^(x)
=
ten

Or if nosotros summate the logarithm of the exponential function of x,

f

^-1(f
(ten)) = log
_b
(b^x
) =
x

Natural logarithm (ln)

Natural logarithm is a logarithm to the base east:

ln(x) = log
_e
(10)

When e abiding is the number:

or

Come across: Natural logarithm

Changed logarithm adding

The inverse logarithm (or anti logarithm) is calculated by raising the base b to the logarithm y:

x
= log^-1(y) =
b
^y

Logarithmic function

The logarithmic part has the basic form of:

f
(x) = log
_b
(x)

Logarithm rules

Rule name	Rule
Logarithm product dominion	log b (x ∙ y) = log b (ten) + log b (y)
Logarithm quotient rule	log b (x / y) = log b (x) – log b (y)
Logarithm power rule	log b (x y ) = y ∙ log b (x)
Logarithm base switch rule	log b (c) = 1 / log c (b)
Logarithm base of operations change dominion	log b (x) = log c (x) / log c (b)
Derivative of logarithm	f (x) = log b (x) ⇒ f ‘ (x) = 1 / ( x ln(b) )
Integral of logarithm	∫ log b (ten) dx = x ∙ ( log b (10) – 1 / ln(b) ) + C
Logarithm of negative number	log b (ten) is undefined when 10≤ 0
Logarithm of 0	log b (0) is undefined
Logarithm of i	log b (1) = 0
Logarithm of the base	log b (b) = one
Logarithm of infinity	lim log b (x) = ∞, when x→∞

Come across: Logarithm rules

Logarithm product rule

The logarithm of the multiplication of x and y is the sum of logarithm of 10 and logarithm of y.

log
_b
(x ∙ y) = log
_b
(10)
+
log
_b
(y)

For example:

log₁₀(3
∙
7) = log₁₀(3)
+
log₁₀(7)

Logarithm quotient rule

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

log
_b
(x / y) = log
_b
(ten)
–
log
_b
(y)

For example:

log_ten(iii
/
seven) = log₁₀(3)
–
log₁₀(7)

Logarithm power rule

The logarithm of x raised to the power of y is y times the logarithm of 10.

log
_b
(x
^y
) =
y ∙
log
_b
(ten)

For example:

log₁₀(two
⁸
) = 8∙
log_x(2)

Logarithm base switch rule

The base b logarithm of c is ane divided by the base c logarithm of b.

log
_b
(c) = 1 / log
_c
(b)

For example:

log_two(8) = 1 / log₈(2)

Logarithm base change rule

The base of operations b logarithm of x is base c logarithm of x divided by the base c logarithm of b.

log
_b
(x) = log
_c
(x) / log
_c
(b)

For example, in gild to calculate log₂(8) in calculator, we need to change the base of operations to 10:

log_ii(8) = log_ten(8) / log_ten(2)

See: log base change rule

Logarithm of negative number

The base b real logarithm of x when 10<=0 is undefined when x is negative or equal to cipher:

log
_b
(x)
is undefined when
x
≤ 0

Meet: log of negative number

Logarithm of 0

The base of operations b logarithm of zero is undefined:

log
_b
(0)
is undefined

The limit of the base of operations b logarithm of x, when x approaches zero, is minus infinity:

See: log of naught

Logarithm of ane

The base b logarithm of one is aught:

log
_b
(1) = 0

For example, teh base ii logarithm of one is naught:

log₂(one) = 0

Run across: log of one

Logarithm of infinity

The limit of the base b logarithm of x, when ten approaches infinity, is equal to infinity:

lim log
_b
(x) = ∞,
when

x→∞

Come across: log of infinity

Logarithm of the base of operations

The base b logarithm of b is one:

log
_b
(b) = i

For example, the base of operations two logarithm of two is one:

log₂(2) = 1

Logarithm derivative

When

f
(x) = log
_b
(x)

Then the derivative of f(x):

f ‘
(x) = 1 / (
ten
ln(b) )

See: log derivative

Logarithm integral

The integral of logarithm of x:

∫
log
_b
(x)
dx
=
ten ∙
( log
_b
(x)
– 1 / ln(b)
) +
C

For example:

∫
log₂(10)
dx
=
x ∙
( log₂(x)
– ane / ln(two)
) +
C

Logarithm approximation

log₂(10) ≈
n
+ (10/two
ⁿ

– ane) ,

Complex logarithm

For complex number z:

z = re^iθ
= ten + iy

The complex logarithm volition be (n = …-2,-1,0,1,two,…):

Log
z =
ln(r) +
i(θ+2nπ)
=
ln(√(x
²+y
²)) +
i·arctan(y/x))

Logarithm bug and answers

Problem #ane

Find x for

log₂(x) + log₂(ten-3) = two

Solution:

Using the product rule:

log₂(x∙(ten-3)) = 2

Changing the logarithm form according to the logarithm definition:

x∙(x-3) = 2^two

Or

ten
ⁱⁱ-3x-4 = 0

Solving the quadratic equation:

x
_1,2
= [iii±√(9+16) ] / 2 = [3±5] / two = four,-1

Since the logarithm is not defined for negative numbers, the answer is:

x
= 4

Trouble #2

Find x for

log_three(x+2) – log_three(x) = 2

Solution:

Using the quotient dominion:

log_three((x+2) /
x) = 2

Changing the logarithm class co-ordinate to the logarithm definition:

(10+ii)/x
= iiiⁱⁱ

Or

10+2 = 9x

Or

8x
= 2

Or

x
= 0.25

Graph of log(x)

log(ten) is not defined for real not positive values of x:

Logarithms table

10	log 10 10	log two x	log east ten
	undefined	undefined	undefined
+	– ∞	– ∞	– ∞
0.0001	-4	-thirteen.287712	-9.210340
0.001	-three	-9.965784	-six.907755
0.01	-two	-six.643856	-4.605170
0.one	-1	-3.321928	-two.302585
1
2	0.301030	1	0.693147
3	0.477121	1.584963	i.098612
4	0.602060	2	1.386294
v	0.698970	2.321928	ane.609438
6	0.778151	2.584963	i.791759
vii	0.845098	2.807355	1.945910
eight	0.903090	3	two.079442
9	0.954243	iii.169925	two.197225
10	i	three.321928	ii.302585
20	1.301030	iv.321928	2.995732
30	1.477121	iv.906891	3.401197
40	1.602060	5.321928	3.688879
50	one.698970	5.643856	iii.912023
60	1.778151	5.906991	4.094345
70	1.845098	6.129283	4.248495
fourscore	one.903090	6.321928	4.382027
90	one.954243	6.491853	4.499810
100	two	six.643856	iv.605170
200	2.301030	7.643856	v.298317
300	2.477121	8.228819	5.703782
400	2.602060	8.643856	five.991465
500	2.698970	8.965784	6.214608
600	2.778151	nine.228819	6.396930
700	2.845098	ix.451211	six.551080
800	2.903090	nine.643856	6.684612
900	two.954243	ix.813781	6.802395
yard	3	9.965784	6.907755
10000	4	xiii.287712	9.210340

Logarithm calculator ►

---

See also

• Logarithm rules
• Logarithm change of base of operations
• Logarithm of zero
• Logarithm of ane
• Logarithm of infinity
• Logarithm of negative number
• Logarithm estimator
• Logarithm graph
• Logarithm tabular array
• Natural logarithm estimator
• Natural logarithm – ln x
• east constant
• Decibel (dB)