#### Binary Logarithm,Natural Logarithm,Common Logarithm,Anti-Logarithm Calculation

# Logarithm Calculator

An online complete log logarithm calculator.

**log**

_{ }

^{}

**ln**

**log**

_{10}**log**

_{2}**Antilog**

_{ }

^{}

**Logarithm** is the **inverse function to exponentiation**.

First, we see about exponentiation,

Exponentiation is an expression that involves two numbers, a base and an exponent, where an exponent is mathematical shorthand representing how many times a number is multiplied against itself.

**Example**

2^{1} = 2

2^{2} = 4

2^{3} = 8

2^{4} = 16

2^{5} = 32

2^{5} = 64

with same above entries, Now we see about Logarithm, i.e __Inverse function to exponentiation__

log_{2} 2 = 1

log_{2} 4 = 2

log_{2} 8 = 3

log_{2} 16 = 4

log_{2} 32 = 5

log_{2} 64 = 6

###### Logarithm Formulas Used In Our Calculators

**1. Log Formula**

log_{b}(x) = y

x = log_{b}(b^{x})

**log _{b}(x) = y** is equivalent to

**x = b**

^{y}(log_{b}(x) => This read as log base b of x is equals to y )

**b: ** log base number, b>0 and b≠1

**x: ** is real number, x>0

**2. Natural Logarithm or Logarithm Base e Formula**

Log base e is also called as natural logarithm.

Natural logarithm symbol is **ln**.

ln(x) = y

**ln(x)** is equivalent to **log _{e}(x)**

**x: ** is real number, x > 0

**3. Common Logarithm or Logarithm Base 10 Formula**

Log base 10 is also called as common logarithm.

log_{10}(x) = y is equivalent to x = 10^{y}

log_{10}(x) = log(x)

**x: ** is real number, x>0

**4. Binary Logarithm or Logarithm Base 2 Formula**

Log base 2 is also called as binary logarithm.

log_{2}(x) = y is equivalent to x = 2^{y}

**x: ** is real number, x>0

**5. Antilogarithm (or Inverse logarithm) Formula**

Calculate the inverse logarithm of a number.

When

*y* = log_{b }x

The anti-logarithm is calculated by raising the base b to the logarithm y

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

## List Of Logarithmic Laws or Rules or Identities

Important formulas, sometimes called logarithmic identities or logarithmic laws.They are

**Logarithm of a Product**

log_{b} (*xy*) = log_{b} *x* + log_{b} *y*

**Example**

log_{3} 243 = log_{3} (9 . 27) = log_{3} 9 + log_{3} 27 = 2 + 3 = 5

**Logarithm of a Quotient**

log_{b} (*x/y*) = log_{b} *x* - log_{b} *y*

**Example**

log_{2} 16 = log_{2} (64/4) = log_{2} 64 - log_{2} 4 = 6 - 2 = 4

**Logarithm of a Power**

log_{b} (*x ^{p}*) =

*p*log

_{b}

*x*

The logarithm of an power number where its base is the same as the base of the log equals the power.

**Example**

log_{2} 64 = log_{2} (2^{6}) = 6 log_{2} 2 = 6

**Logarithm of a Root**

log_{b} ^{p}√*x* = (log_{b} *x*) / *p*

**Example**

log_{10} √1000 = (1 / 2) . log_{10} 1000 = 3 / 2 = 1.5

**Logarithm of Zero**

log_{b} (1) = 0

The logarithm of 1 with b > 1 equals zero.

**Logarithm of Identity**

log_{b} (*b*) = 1

The logarithm of a number that is equal to its base is just 1.

**Logarithm of Exponent**

*b*^{logb (k)} = k

Raising the logarithm of a number by its base equals the number.

**Change of Base**

log_{b} (*x*) = (log_{k} (*x*)) / (log_{k} (*b*))

## Common Values for Log Base b

Base b | Name for log_{b}x | ISO notation | Other notations |
---|---|---|---|

2 | Binary logarithm | lb x | ld x, log x, lg x, log2x |

e | Natural logarithm | ln x | log x |

10 | Common logarithm | lg x | log x, log10x |

## Logarithm Values Tables

log_{b}(x) = y |
---|

log_{2} (1) = 0 |

log_{2} (2) = 1 |

log_{2} (3) = 1.584962501 |

log_{2} (4) = 2 |

log_{2} (5) = 2.321928095 |

log_{2} (6) = 2.584962501 |

log_{2} (7) = 2.807354922 |

log_{2} (8) = 3 |

log_{2} (9) = 3.169925001 |

log_{2} (10) = 3.321928095 |

log_{2} (11) = 3.459431619 |

log_{2} (12) = 3.584962501 |

log_{2} (13) = 3.700439718 |

log_{2} (14) = 3.807354922 |

log_{2} (15) = 3.906890596 |

log_{2} (16) = 4 |

log_{2} (17) = 4.087462841 |

log_{2} (18) = 4.169925001 |

log_{2} (19) = 4.247927513 |

log_{2} (20) = 4.321928095 |

log_{2} (21) = 4.392317423 |

log_{2} (22) = 4.459431619 |

log_{2} (23) = 4.523561956 |

log_{2} (24) = 4.584962501 |

log_{2} (25) = 4.64385619 |

log_{2} (26) = 4.700439718 |

log_{2} (27) = 4.754887502 |

log_{2} (28) = 4.807354922 |

log_{2} (29) = 4.857980995 |

log_{2} (30) = 4.906890596 |

log_{2} (31) = 4.95419631 |

log_{2} (32) = 5 |

log_{2} (33) = 5.044394119 |

log_{2} (34) = 5.087462841 |

log_{2} (35) = 5.129283017 |

log_{2} (36) = 5.169925001 |

log_{2} (37) = 5.209453366 |

log_{2} (38) = 5.247927513 |

log_{2} (39) = 5.285402219 |

log_{2} (40) = 5.321928095 |

log_{2} (41) = 5.357552005 |

log_{2} (42) = 5.392317423 |

log_{2} (43) = 5.426264755 |

log_{2} (44) = 5.459431619 |

log_{2} (45) = 5.491853096 |

log_{2} (46) = 5.523561956 |

log_{2} (47) = 5.554588852 |

log_{2} (48) = 5.584962501 |

log_{2} (49) = 5.614709844 |

log_{2} (50) = 5.64385619 |