# Logarithm and Anti-Log Calculator

## Logarithm

**Logarithm** is a basic math function used to find how many times log base should be multiplied to produce a given number. Logarithm of **x** to base **b** equals to **y** can be mathematically described as **log _{b} x = y** which defines that the base "b" multiplied "y" times to produce a given number "x".

The below is some of the arithmetic rules and properties of logarithm function for multiplication, division, power and root.

Log and Anti-Log Formula | ||
---|---|---|

Operation | Formula | Example |

Multilication | log_{b}(x*y) = log_{b} x + log_{b} y | log_{5}(75) = log_{5}(3 * 25) = log_{5}(3) + log_{5}(25) |

Division | log_{b}(x/y) = log_{b}(x) - log_{b}(y) | log_{2}(1/4) = log_{2}(1) - log_{2}(4) = 0 - 2 = -2 |

Power | log_{b}(x^{y}) = y * log_{b}(x) | log_{2}(2^{4}) = 4 * log_{2}(2) = 4 |

Logarithm Table for Base 2 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

# | .0 | .1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |

0 | -∞ | -3.3219 | -2.3219 | -1.737 | -1.3219 | -1 | -0.737 | -0.5146 | -0.3219 | -0.152 |

1 | 0 | 0.1375 | 0.263 | 0.3785 | 0.4854 | 0.585 | 0.6781 | 0.7655 | 0.848 | 0.926 |

2 | 1 | 1.0704 | 1.1375 | 1.2016 | 1.263 | 1.3219 | 1.3785 | 1.433 | 1.4854 | 1.5361 |

3 | 1.585 | 1.6323 | 1.6781 | 1.7225 | 1.7655 | 1.8074 | 1.848 | 1.8875 | 1.926 | 1.9635 |

4 | 2 | 2.0356 | 2.0704 | 2.1043 | 2.1375 | 2.1699 | 2.2016 | 2.2327 | 2.263 | 2.2928 |

5 | 2.3219 | 2.3505 | 2.3785 | 2.406 | 2.433 | 2.4594 | 2.4854 | 2.511 | 2.5361 | 2.5607 |

6 | 2.585 | 2.6088 | 2.6323 | 2.6554 | 2.6781 | 2.7004 | 2.7225 | 2.7442 | 2.7655 | 2.7866 |

7 | 2.8074 | 2.8278 | 2.848 | 2.8679 | 2.8875 | 2.9069 | 2.926 | 2.9449 | 2.9635 | 2.9819 |

8 | 3 | 3.0179 | 3.0356 | 3.0531 | 3.0704 | 3.0875 | 3.1043 | 3.121 | 3.1375 | 3.1538 |

9 | 3.1699 | 3.1859 | 3.2016 | 3.2172 | 3.2327 | 3.2479 | 3.263 | 3.278 | 3.2928 | 3.3074 |

10 | 3.3219 | 3.3363 | 3.3505 | 3.3646 | 3.3785 | 3.3923 | 3.406 | 3.4195 | 3.433 | 3.4463 |

Logarithm Table for Base 10 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

# | .0 | .1 | .2 | .3 | .4 | .5 | .6 | .7 | .8 | .9 |

0 | -∞ | -1 | -0.699 | -0.5229 | -0.3979 | -0.301 | -0.2218 | -0.1549 | -0.0969 | -0.0458 |

1 | 0 | 0.0414 | 0.0792 | 0.1139 | 0.1461 | 0.1761 | 0.2041 | 0.2304 | 0.2553 | 0.2788 |

2 | 0.301 | 0.3222 | 0.3424 | 0.3617 | 0.3802 | 0.3979 | 0.415 | 0.4314 | 0.4472 | 0.4624 |

3 | 0.4771 | 0.4914 | 0.5051 | 0.5185 | 0.5315 | 0.5441 | 0.5563 | 0.5682 | 0.5798 | 0.5911 |

4 | 0.6021 | 0.6128 | 0.6232 | 0.6335 | 0.6435 | 0.6532 | 0.6628 | 0.6721 | 0.6812 | 0.6902 |

5 | 0.699 | 0.7076 | 0.716 | 0.7243 | 0.7324 | 0.7404 | 0.7482 | 0.7559 | 0.7634 | 0.7709 |

6 | 0.7782 | 0.7853 | 0.7924 | 0.7993 | 0.8062 | 0.8129 | 0.8195 | 0.8261 | 0.8325 | 0.8388 |

7 | 0.8451 | 0.8513 | 0.8573 | 0.8633 | 0.8692 | 0.8751 | 0.8808 | 0.8865 | 0.8921 | 0.8976 |

8 | 0.9031 | 0.9085 | 0.9138 | 0.9191 | 0.9243 | 0.9294 | 0.9345 | 0.9395 | 0.9445 | 0.9494 |

9 | 0.9542 | 0.959 | 0.9638 | 0.9685 | 0.9731 | 0.9777 | 0.9823 | 0.9868 | 0.9912 | 0.9956 |

10 | 1 | 1.0043 | 1.0086 | 1.0128 | 1.017 | 1.0212 | 1.0253 | 1.0294 | 1.0334 | 1.0374 |

**Solved Example Problem for Logarithm**

The below solved example problem may help you understand the mathematical function of logarithm. Use this logarithm calculator to generate steps to find base-2, base-10 or natural logarithm for any given number.

__Example Problem:__

log_{2}(1/64) = ?

What is log base 2 of 1/64?

__Workout :__

step 1 Address the formula, input parameters and values

Formula:

log_{b}(x) = y, if b^{y} = x

`x` = 1/64

`b` = 2

log_{2}(1/64) = y

step 2 Write the number 1/64 in raising 2 to the nth power

(2)^{y} = 1/64

1/64 = 1/2^{6}

1/64 = 2^{-6}

(2)^{y} = 2^{-6}

y = -6

log_{2}(1/64) = -6

## Inverse Logarithm

**Anti-log** or inverse logarithm function is also a basic math function used to find the value of exponential function. In mathematics, an inverse log of 3 to the base 10 mathematically represented by 10^{y} = x.

**Solved Example Problem for Inverse Logarithm**
The below solved example problem may help you understand the mathematical function of anti-log or inverse logarithm. Use this antilog calculator to generate steps to find inverse logarithm for any given number.

__Example Problem :__

antilog_{2}(6) = ?

What is antilog (base 10) of 4?__Workout :__

step 1 Address the formula input parameters and values

Formula:

log_{b}(x) = y, if b^{y} = x
log_{10}x = 4

y = 4

b = 10

step 2 Find the antilog value as per the formula

From the above formula

10^{4} = x

x = 10000

log_{10}(10000) = 4