The online log calculator allows you to calculate the logarithm of any number with the chosen base. Our free equation logarithm calculator shows you fast and step-by-step calculations with 100% accurate results.

## What Is a Logarithm?

**A logarithm is a power through which the number is raised in order to get some other numbers.**

In other words, it is the inverse of the exponent function.

**For Example:**

If the logarithm of 1000 with base 10 is 3. The logarithm of 1000 is 3 and how many times 10 needs to be multiplied by itself.

## Types of a Logarithm:

The log base calculator allows you to compute math log of numbers with arbitrary bases.

The types it deals with include:

**Common Logarithm:**

The power to which a base of 10 must be raised to obtain is called the common logarithm.

It is conventionally denoted as lg(x). It is also known as the decimal logarithm.

**Natural Logarithm:**

The power to which a base of special number e must be raised to obtain is called the natural logarithm.

The log of e value is approximately equal to 2.718281828459.

## Logarithm Rules Table:

The single logarithmic calculator gives you rapid calculations of multiplication, division, subtraction, addition, squares, and roots. The rules are below.

Rule | Formula | Intension |
---|---|---|

Zero Rule | Logb 1 = 0 | The logarithm of the 1 to any base is always equal to the zero. b is positive but b = 1. |

Base Rule | Logb b = 1 | The logarithm of the argument wherein the argument equals the base = 1 |

Product Rule |
Logb (mn) = logb m + logb n | The logarithm of the product is the sum of the logarithms of the factors. |

Quotient rule | Logb m/n = logb m - logb n | The logarithm of the ratio of the two quantities is the logarithm of the numerator - the logarithm of the denominator. |

Power rule | Logb m^n = n logb m | The logarithm of the exponential number is the exponent times the logarithm of the base. |

Inverse property of log | b^logbx = x | The logarithm of the exponential number where its base is the same as the base of the log is equal to the exponent. |

Change of base rule | Loga b = logc b/logc a OR Loga b . Logc a = logc b | The logarithm of the base on the left side is equal to the base on the right side are equals to the c |

Equality rule | Logb a = logb c → a = c | The logarithm of the equality rule equals the log b which means that base a equals the base c |

How To Calculate Logarithms?

The log calculator computes the exponent and power. Let’s get fast and exact solutions in the form of an example.

### Example:

Calculate the math log and finds which rule is most suitable for the following expression.

**Log (5) + Log (2)**

#### Solution:

Use the product property of the logarithms calculator that is:

Logb (mn) = logb m + logb n

Log ( 5 . 2) = log 5 + log 2

Log (10) = 0.6990 + 0.3010

Log 10 = 1

1 = 1

So we say that the logarithm base 10 of 10 is 1. For verifications, you may use this log properties calculator to enter the number and get accurate log computations. For antilog calculation, you may use our __antilog calculator__.

## Working of Log Calculator:

The basic properties of the logarithms calculator are calculated by our free tool. Just follow the following steps to make your calculations fast and accurate.

**Input:**

- Choose the nature of the logarithm
- Enter the number & base in their designated fields
- Tap
**“calculate”**

**Output:**

- Log value
- Step-by-step calculations

## FAQs:

### What Is The Relation Between The Exponent And Logarithm?

The exponent is the reverse function of the logarithm and vice versa. It is clear that the log and inverse log are both other terms for the exponent.

### What Is The Value Of Log10?

Log 10 is the logarithmic function of base 10. The value of log10 is always equal to 1. While the loge 10 = 2.302585

### What Are The Applications Of The Log With Bases?

**Binary logarithm:**

It is the log with base 2 and is used in computer science, information theory, bioinformatics, music theory, and in photography.

**Natural logarithm:**

It is a log base e and is used in mathematics, chemistry, statistics, economics, information theory, and in engineering.

**Common logarithm:**

It is a log base 10 and is used in various engineering fields, logarithm tables, handheld calculations, and spectroscopy.

## Logarithm Table:

Table of base 10, base 2, and base e (ln) logarithms:

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 |

## References:

From the source Wikipedia: Logarithm, Definition, Logarihtm identities, particular bases, History, logarithm tables, slide rules, and applications, analytic properties.