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:

log* _{b }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* = log_{b }x

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

*x* = log_{b}^{-1}(*y*) = *b ^{ y}*

- Logarithm definition
- Logarithm rules
- Logarithm problems
- Complex logarithm
- Graph of log(x)
- Logarithm table
- Logarithm calculator

## 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:

log* _{b}*(

*x*)

*= y*

For example when:

2^{4} = 16

Then

log_{2}(16) = 4

## Logarithm as inverse function of exponential function

The logarithmic function,

*y *= log* _{b}*(

*x*)

is the inverse function of the exponential function,

*x *=* b ^{y}*

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

*f *(*f *^{-1}(*x*)) = *b*^{log}*b*^{(x)} = *x*

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

*f *^{-1}(*f *(*x*)) = log_{b}(*b ^{x}*) =

*x*

## Natural logarithm (ln)

Natural logarithm is a logarithm to the base e:

ln(*x*) = log* _{e}*(

*x*)

When e constant is the number:

or

See: Natural logarithm

## 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:

*f *(*x*) = log* _{b}*(

*x*)

## Logarithm rules

Rule name | Rule |
---|---|

## Logarithm product rule | log(_{b}x ∙ y) = log(_{b}x) + 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 change rule | 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}x) dx = x ∙ ( 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}(1) = 0 |

## Logarithm of the base | log_{b}(b) = 1 |

## Logarithm of infinity | lim log_{b}(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.

log* _{b}*(

*x ∙ y*) = log

*(*

_{b}*x*)

*+*log

*(*

_{b}*y*)

#### For example:

log_{10}(3* ∙ *7) = log_{10}(3)* + *log_{10}(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}*x*)

*-*log

*(*

_{b}*y*)

#### For example:

log_{10}(3* / *7) = log_{10}(3)* - *log_{10}(7)

#### Logarithm power rule

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

log* _{b}*(

*x*) =

^{y}*y ∙*log

*(*

_{b}*x*)

#### For example:

log_{10}(2^{8}) = 8*∙ *log_{10}(2)

#### Logarithm base switch rule

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

log* _{b}*(

*c*) = 1 / log

*(*

_{c}*b*)

#### For example:

log_{2}(8) = 1 / log_{8}(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.

log* _{b}*(

*x*) = log

*(*

_{c}*x*) / log

*(*

_{c}*b*)

For example, in order to calculate log_{2}(8) in calculator, we need to change the base to 10:

log_{2}(8) = log_{10}(8) / log_{10}(2)

See: log base change rule

#### Logarithm of negative number

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

log_{b}(*x*) is undefined when *x* ≤ 0

#### Logarithm of 0

The base b logarithm of zero is undefined:

log_{b}(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:

log_{b}(1) = 0

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

log_{2}(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 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 one:

log_{2}(2) = 1

#### Logarithm derivative

When

*f *(*x*) = log* _{b}*(

*x*)

Then the derivative of f(x):

*f ' *(*x*) = 1 / (* x* ln(*b*) )

See: log derivative

#### Logarithm integral

The integral of logarithm of x:

∫* *log* _{b}*(

*x*)

*dx*=

*x ∙*( log

*(*

_{b}*x*)

*- 1 / ln(*

*b*)

*) +*

*C*

#### For example:

∫* *log_{2}(*x*) *dx* = *x ∙ *( log_{2}(*x*)* *- 1 / ln(2)* *) + *C*

## Logarithm approximation

log_{2}(*x*) ≈ *n* + (*x*/2^{n} - 1) ,

## Complex logarithm

For complex number z:

*z = re ^{iθ} = x + iy*

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

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

## Logarithm problems and answers

#### Problem #1

Find x for

log_{2}(*x*) + log_{2}(*x*-3) = 2

#### Solution:

Using the product rule:

log_{2}(*x∙*(*x*-3)) = 2

Changing the logarithm form according to the logarithm definition:

*x∙*(*x*-3) = 2^{2}

Or

*x*^{2}-3*x*-4 = 0

Solving the quadratic equation:

*x*_{1,2} = [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

log_{3}(*x*+2) - log_{3}(*x*) = 2

#### Solution:

Using the quotient rule:

log_{3}((*x*+2) /* x*) = 2

Changing the logarithm form according to the logarithm definition:

(*x*+2)/*x* = 3^{2}

Or

*x*+2 = 9*x*

Or

8*x* = 2

Or

*x* = 0.25

## Graph of log(x)

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

## Logarithms table

x | log_{10}x | log_{2}x | log_{e}x |
---|---|---|---|

0 | undefined | undefined | undefined |

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 |

1 | 0 | 0 | 0 |

2 | 0.301030 | 1 | 0.693147 |

3 | 0.477121 | 1.584963 | 1.098612 |

4 | 0.602060 | 2 | 1.386294 |

5 | 0.698970 | 2.321928 | 1.609438 |

6 | 0.778151 | 2.584963 | 1.791759 |

7 | 0.845098 | 2.807355 | 1.945910 |

8 | 0.903090 | 3 | 2.079442 |

9 | 0.954243 | 3.169925 | 2.197225 |

10 | 1 | 3.321928 | 2.302585 |

20 | 1.301030 | 4.321928 | 2.995732 |

30 | 1.477121 | 4.906891 | 3.401197 |

40 | 1.602060 | 5.321928 | 3.688879 |

50 | 1.698970 | 5.643856 | 3.912023 |

60 | 1.778151 | 5.906991 | 4.094345 |

70 | 1.845098 | 6.129283 | 4.248495 |

80 | 1.903090 | 6.321928 | 4.382027 |

90 | 1.954243 | 6.491853 | 4.499810 |

100 | 2 | 6.643856 | 4.605170 |

200 | 2.301030 | 7.643856 | 5.298317 |

300 | 2.477121 | 8.228819 | 5.703782 |

400 | 2.602060 | 8.643856 | 5.991465 |

500 | 2.698970 | 8.965784 | 6.214608 |

600 | 2.778151 | 9.228819 | 6.396930 |

700 | 2.845098 | 9.451211 | 6.551080 |

800 | 2.903090 | 9.643856 | 6.684612 |

900 | 2.954243 | 9.813781 | 6.802395 |

1000 | 3 | 9.965784 | 6.907755 |

10000 | 4 | 13.287712 | 9.210340 |

Currently, we have around 940 calculators, conversion tables and usefull online tools and software features for students, teaching and teachers, designers and simply for everyone.

You can find at this page financial calculators, mortgage calculators, calculators for loans, calculators for auto loan and lease calculators, interest calculators, payment calculators, retirement calculators, amortization calculators, investment calculators, inflation calculators, finance calculators, income tax calculators, compound interest calculators, salary calculator, interest rate calculator, sales tax calculator, fitness & health calculators, bmi calculator, calorie calculators, body fat calculator, bmr calculator, ideal weight calculator, pace calculator, pregnancy calculator, pregnancy conception calculator, due date calculator, math calculators, scientific calculator, fraction calculator, percentage calculators, random number generator, triangle calculator, standard deviation calculator, other calculators, age calculator, date calculator, time calculator, hours calculator, gpa calculator, grade calculator, concrete calculator, subnet calculator, password generator conversion calculator and many other tools and for text editing and formating, downloading videos from Facebok (we built one of the most famous Facebook video downloader online tools). We also provide you online downloanders for YouTube, Linkedin, Instagram, Twitter, Snapchat, TikTok and other social media sites (please note we does not host any videos on its servers. All videos that you download are downloaded from Facebook's, YouTube's, Linkedin's, Instagram's, Twitter's, Snapchat's, TikTok's CDNs. We also specialise on keyboard shortcuts, ALT codes for Mac, Windows and Linux and other usefull hints and tools (how to write emoji online etc.)

There are many very usefull online free tools and we would be happy if you share our page to others or send us any suggestions for other tools which will come on your mind. Also in case you find any of our tools that it does not work properly or need a better translation - please let us know. Our tools will make your life easier or simply help you to do your work or duties faster and in more effective way.

These below are the most commonly used by many users all over the world.

- Free online calculators and tools
- Free online units conversion tools
- Free online web design tools
- Free online electricity & electronics tools
- Mathematics
- Online Tools
- Text Tools
- PDF Tools
- Code
- Ecology
- Free online electrical calculators and tools
- Free online financial calculators and tools
- Free online grade calculators and tools
- Free online lighting calculators and tools
- Free online math calculators and tools
- Free online wire calculators and tools
- Free online time calculators and tools
- Free online baby calculators and tools
- Free online body calculators and tools
- Math free online calculator
- Adding fractions free online calculator
- Addition free online calculator
- Antilog free online calculator
- Arccos free online calculator
- Arcsin free online calculator
- Arctan free online calculator
- Average free online calculator
- Base free online calculator
- Binary free online calculator
- Convolution free online calculator
- Cosine free online calculator
- Dividing fractions free online calculator
- Division free online calculator
- Exponential growth/decay formula calculator online free
- Exponents free online calculator
- Factorial free online calculator
- Fractions free online calculator
- Greatest common factor free online calculator
- Least common multiple free online calculator
- Natural logarithm free online calculator
- Logarithm free online calculator
- Multiplication free online calculator
- Multiplying fractions free online calculator
- Percentage free online calculator
- Percentage change free online calculator
- Percent error free online calculator
- Percentage increase/decrease free online calculator
- Pythagorean theorem free online calculator
- Quadratic equation free online calculator
- Random number free online generator
- Ratio free online calculator
- Radicals and roots free online calculator
- Scientific notation free online calculator
- Simplifying fractions free online calculator
- Sine free online calculator
- Square root free online calculator
- Standard deviation free online calculator
- Subtracting fractions free online calculator
- Subtraction free online calculator
- Tangent free online calculator
- Trigonometry free online calculator
- Weighted average free online calculator
- Variance free online calculator
- Random number generator 1-2
- Random number generator 1-3
- Random number generator 1-4
- Random number generator 1-5
- Random number generator 1-6
- Random number generator 1-10
- Random number generator 1-20
- Random number generator 1-100
- Random number generator 0-1
- Random number generator 0-9
- Random number generator 0-10
- Random number generator 0-99
- Random number generator 0-100

And we are still developing more. Our goal is to become the one-stop, go-to site for people who need to make quick calculations or who need to find quick answer for basic conversions.

Additionally, we believe the internet should be a source of free information. Therefore, all of our tools and services are completely free, with no registration required. We coded and developed each calculator individually and put each one through strict, comprehensive testing. However, please inform us if you notice even the slightest error – your input is extremely valuable to us. While most calculators on Justfreetools.com are designed to be universally applicable for worldwide usage, some are for specific countries only.