# Log 0 001

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:

2iv
= 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
=
by

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

log
b
(y)
log
b
(x
y
) =
y ∙
log
b
(x)
log
b
(c) = 1 / log
c
(b)
log
b
(x) = log
c
(x) / log
c
(b)
f
(x) = log
b
(x)

f ‘
(x) = 1 / (
x

ln(b) )

log
b
(ten)
dx
=
x ∙
( log
b
(10)
– 1 / ln(b)
) +
C
##### Logarithm of negative number
log
b
(ten)
is undefined when

10≤ 0
log
b
(0)
is undefined
log
b
(1) = 0
log
b
(b) = one
##### Logarithm of infinity
lim log
b
(x) =
∞,
when

x
→∞
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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:

logten(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
logx(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:

logtwo(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:

logii(8) = logten(8) / logten(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:

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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

log2(10) ≈
n
+ (10/two
n

– ane) ,

## Complex logarithm

For complex number z:

z = re
= 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) = 2two

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

logthree(x+2) – logthree(x) = 2

##### Solution:

Using the quotient dominion:

logthree((x+2) /
x
) = 2

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

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(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 ►

• 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)

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### Log 0 001

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