# Log 10 X 2 Log 2x 10

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

logtwo(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
=
by

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,

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f

-one(f
(ten)) = log
b
(bten
) =
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

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
log
b
(ten ∙ y) = log
b
(10)
+
log
b
(y)
log
b
(x / y) = log
b
(x)

log
b
(y)
log
b
(10
y
) =
y ∙
log
b
(x)
log
b
(c) = i / log
c
(b)
log
b
(x) = log
c
(10) / log
c
(b)
f
(ten) = log
b
(10)

f ‘
(x) = 1 / (
x

ln(b) )

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 log
b
(ane) = 0
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) = logx(3)
+
logx(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:

logii(8) = 1 / log8(2)

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#### 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) = logx(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:

logii(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

log2(x) ≈
due north
+ (10/2
north

– 1) ,

## Complex logarithm

For circuitous number z:

z = re
= 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))

#### Problem #one

Find x for

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logii(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

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

logiii(x+two) – log3(10) = 2

##### Solution:

Using the quotient rule:

logthree((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 ►

## 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)