Log 10 X 2 Log 2x 10

Logarithm Rules

The
base
b
logarithm
of a number is the
exponent
that we need to raise the
base
in order to get the number.

• Logarithm definition
• Logarithm rules
• Logarithm problems
• Complex logarithm
• Graph of log(ten)
• Logarithm tabular array
• Logarithm estimator

Logarithm definition

When b is raised to the ability of y is equal x:

b
^y

=
x

Then the base of operations b logarithm of x is equal to y:

log
_b
(x)
= y

For example when:

2⁴
= 16

Then

log_two(sixteen) = 4

Logarithm as inverse function of exponential function

The logarithmic office,

y
= log
_b
(x)

is the inverse function of the exponential role,

x
=
b^y

So if nosotros calculate the exponential function of the logarithm of x (x>0),

f
(f

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

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

f

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

Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(x) = log
_e
(10)

When e constant is the number:

or

Come across: Natural logarithm

Inverse 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 function has the basic form of:

f
(x) = log
_b
(x)

Logarithm rules

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

See: Logarithm rules

Logarithm product dominion

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

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

For case:

log₁₀(3
∙
7) = log_x(3)
+
log_x(7)

Logarithm caliber rule

The logarithm of the segmentation of x and y is the difference of logarithm of 10 and logarithm of y.

log
_b
(x / y) = log
_b
(10)
–
log
_b
(y)

For example:

log₁₀(three
/
7) = log₁₀(three)
–
log₁₀(7)

Logarithm power rule

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

log
_b
(x
^y
) =
y ∙
log
_b
(x)

For case:

log₁₀(2
⁸
) = eight∙
log₁₀(two)

Logarithm base of operations switch dominion

The base b logarithm of c is 1 divided by the base c logarithm of b.

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

For instance:

log_ii(8) = 1 / log₈(2)

Logarithm base change rule

The base b logarithm of 10 is base c logarithm of ten divided by the base c logarithm of b.

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

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

log₂(viii) = log_x(eight) / log₁₀(ii)

See: log base of operations change rule

Logarithm of negative number

The base of operations b real logarithm of x when ten<=0 is undefined when x is negative or equal to zero:

log
_b
(x)
is undefined when
x
≤ 0

Run across: 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:

Encounter: log of cipher

Logarithm of 1

The base b logarithm of ane is nada:

log
_b
(one) = 0

For example, teh base two logarithm of 1 is zero:

log₂(1) = 0

Meet: log of 1

Logarithm of infinity

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

lim log
_b
(x) = ∞,
when

x→∞

See: log of infinity

Logarithm of the base

The base b logarithm of b is one:

log
_b
(b) = 1

For example, the base two logarithm of two is 1:

log_ii(2) = 1

Logarithm derivative

When

f
(10) = log
_b
(x)

Then the derivative of f(x):

f ‘
(10) = 1 / (
x
ln(b) )

See: log derivative

Logarithm integral

The integral of logarithm of x:

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

For example:

∫
log₂(x)
dx
=
x ∙
( log₂(x)
– ane / ln(2)
) +
C

Logarithm approximation

log₂(x) ≈
due north
+ (10/2
^north

– 1) ,

Complex logarithm

For circuitous number z:

z = re^iθ
= 10 + iy

The complex logarithm volition exist (north = …-2,-i,0,1,two,…):

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

Logarithm bug and answers

Problem #one

Find x for

log_ii(10) + log₂(x-3) = 2

Solution:

Using the product rule:

log₂(x∙(x-3)) = 2

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

x∙(x-three) = 2²

Or

ten
²-3ten-iv = 0

Solving the quadratic equation:

ten
_one,2
= [three±√(ix+16) ] / 2 = [3±5] / two = 4,-1

Since the logarithm is non divers for negative numbers, the answer is:

x
= 4

Trouble #2

Detect x for

log_iii(x+two) – log₃(10) = 2

Solution:

Using the quotient rule:

log_three((ten+2) /
10) = 2

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

(x+2)/ten
= 3ⁱⁱ

Or

x+2 = 9x

Or

8x
= 2

Or

x
= 0.25

Graph of log(x)

log(x) is non defined for real non positive values of x:

Logarithms table

x	log x x	log 2 ten	log e x
	undefined	undefined	undefined
+	– ∞	– ∞	– ∞
0.0001	-four	-xiii.287712	-9.210340
0.001	-3	-9.965784	-six.907755
0.01	-2	-half dozen.643856	-4.605170
0.one	-one	-iii.321928	-2.302585
1
2	0.301030	1	0.693147
iii	0.477121	1.584963	1.098612
four	0.602060	2	1.386294
5	0.698970	ii.321928	i.609438
six	0.778151	2.584963	1.791759
7	0.845098	2.807355	1.945910
8	0.903090	3	2.079442
9	0.954243	three.169925	2.197225
10	1	3.321928	2.302585
20	ane.301030	four.321928	two.995732
30	1.477121	four.906891	3.401197
forty	1.602060	5.321928	3.688879
fifty	1.698970	v.643856	iii.912023
60	1.778151	five.906991	4.094345
lxx	1.845098	6.129283	4.248495
lxxx	1.903090	6.321928	four.382027
90	1.954243	6.491853	four.499810
100	ii	vi.643856	4.605170
200	2.301030	seven.643856	five.298317
300	ii.477121	8.228819	v.703782
400	two.602060	8.643856	v.991465
500	two.698970	eight.965784	6.214608
600	2.778151	9.228819	6.396930
700	2.845098	9.451211	6.551080
800	2.903090	ix.643856	half-dozen.684612
900	ii.954243	9.813781	6.802395
k	3	ix.965784	vi.907755
10000	4	thirteen.287712	9.210340

Logarithm reckoner ►

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See besides

• Logarithm rules
• Logarithm alter of base
• Logarithm of cypher
• Logarithm of i
• Logarithm of infinity
• Logarithm of negative number
• Logarithm calculator
• Logarithm graph
• Logarithm table
• Natural logarithm calculator
• Natural logarithm – ln x
• e constant
• Decibel (dB)