# Logarithm free online calculator

Calculate logarithm of a number to any base:

* Use e for scientific notation. E.g: 5e3, 4e-8, 1.45e12

When:

b y = x

Then the base b logarithm of a number x:

logb x = y

## Anti-logarithm calculator

In order to calculate log-1(y) on the calculator, enter the base b (10 is the default value, enter e for e constant), enter the logarithm value y and press the calculate button:

When

y = logb x

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

x = logb-1(y) = b y

## Logarithm definition

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

b y = x

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

logb(x) = y

For example when:

24 = 16

Then

log2(16) = 4

## Logarithm as inverse function of exponential function

The logarithmic function,

= logb(x)

is the inverse function of the exponential function,

= by

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

(-1(x)) = blogb(x) = x

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

-1((x)) = logb(bx) = x

## Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(x) = loge(x)

When e constant is the number:

or

## Inverse logarithm calculation

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:

(x) = logb(x)

## Logarithm rules

Rule nameRule

#### Logarithm product rule

logb(x ∙ y) = logb(x) + logb(y)

#### Logarithm quotient rule

logb(x / y) = logb(x) - logb(y)

#### Logarithm power rule

logb(y) = y ∙ logb(x)

#### Logarithm base switch rule

logb(c) = 1 / logc(b)

#### Logarithm base change rule

logb(x) = logc(x) / logc(b)

#### Derivative of logarithm

(x) = logb(x) ⇒ f ' (x) = 1 / ( x ln(b) )

#### Integral of logarithm

logb(xdx = x ∙ ( logb(x) - 1 / ln(b) ) + C

#### Logarithm of negative number

logb(x) is undefined when x≤ 0

#### Logarithm of 0

logb(0) is undefined

logb(1) = 0

logb(b) = 1

#### Logarithm of infinity

lim logb(x) = ∞,when x→∞

See: Logarithm rules

#### Logarithm product rule

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

logb(x ∙ y) = logb(x) + logb(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.

logb(x / y) = logb(x) - logb(y)

#### For example:

log10(3 / 7) = log10(3) - log10(7)

#### Logarithm power rule

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

logb(y) = y ∙ logb(x)

#### For example:

log10(28) = 8∙ log10(2)

#### Logarithm base switch rule

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

logb(c) = 1 / logc(b)

#### For example:

log2(8) = 1 / log8(2)

#### Logarithm base change rule

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

logb(x) = logc(x) / logc(b)

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

log2(8) = log10(8) / log10(2)

#### Logarithm of negative number

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

logb(xis undefined when x ≤ 0

#### Logarithm of 0

The base b logarithm of zero is undefined:

logb(0) is undefined

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

See: log of zero

#### Logarithm of 1

The base b logarithm of one is zero:

logb(1) = 0

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

log2(1) = 0

See: log of one

#### Logarithm of infinity

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

lim logb(x) = ∞, when x→∞

See: log of infinity

#### Logarithm of the base

The base b logarithm of b is one:

logb(b) = 1

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

log2(2) = 1

#### Logarithm derivative

When

(x) = logb(x)

Then the derivative of f(x):

f ' (x) = 1 / ( x ln(b) )

See: log derivative

#### Logarithm integral

The integral of logarithm of x:

logb(xdx = x ∙ ( logb(x) - 1 / ln(b) ) + C

#### For example:

log2(xdx = x ∙ ( log2(x) - 1 / ln(2) ) + C

## Logarithm approximation

log2(x) ≈ n + (x/2n - 1) ,

## Complex logarithm

For complex number z:

z = re = x + iy

The complex logarithm will be (n = ...-2,-1,0,1,2,...):

Log z = ln(r) + i(θ+2nπ) = ln(√(x2+y2)) + i·arctan(y/x))

## Logarithm problems and answers

#### Problem #1

Find x for

log2(x) + log2(x-3) = 2

#### Solution:

Using the product rule:

log2(x∙(x-3)) = 2

Changing the logarithm form according to the logarithm definition:

x∙(x-3) = 22

Or

x2-3x-4 = 0

Solving the quadratic equation:

x1,2 = [3±√(9+16) ] / 2 = [3±5] / 2 = 4,-1

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

x = 4

#### Problem #2

Find x for

log3(x+2) - log3(x) = 2

#### Solution:

Using the quotient rule:

log3((x+2) / x) = 2

Changing the logarithm form according to the logarithm definition:

(x+2)/x = 32

Or

x+2 = 9x

Or

8x = 2

Or

x = 0.25

## Graph of log(x)

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

## Logarithms table

xlog10xlog2xlogex
0undefinedundefinedundefined
0+- ∞- ∞- ∞
0.0001-4-13.287712-9.210340
0.001-3-9.965784-6.907755
0.01-2-6.643856-4.605170
0.1-1-3.321928-2.302585
1000
20.30103010.693147
30.4771211.5849631.098612
40.60206021.386294
50.6989702.3219281.609438
60.7781512.5849631.791759
70.8450982.8073551.945910
80.90309032.079442
90.9542433.1699252.197225
1013.3219282.302585
201.3010304.3219282.995732
301.4771214.9068913.401197
401.6020605.3219283.688879
501.6989705.6438563.912023
601.7781515.9069914.094345
701.8450986.1292834.248495
801.9030906.3219284.382027
901.9542436.4918534.499810
10026.6438564.605170
2002.3010307.6438565.298317
3002.4771218.2288195.703782
4002.6020608.6438565.991465
5002.6989708.9657846.214608
6002.7781519.2288196.396930
7002.8450989.4512116.551080
8002.9030909.6438566.684612
9002.9542439.8137816.802395
100039.9657846.907755
10000413.2877129.210340

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