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This online calculator is currently under heavy development. It may or it may NOT work correctly.

You CAN try to use it. You CAN even get the proper results.

However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.

Feel free to send any ideas and comments !

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# Logarithms: general formulas

Name | Formula |

Logarithm of one | Show source$\log_{a} 1 = 0$ |

Logarithm of the product | Show source$\log_{a} \left(x \cdot y \right) = \log_{a} x + \log_{a} y$ |

Logarithm of the quotient | Show source$\log_{a} \frac{x}{y} = \log_{a} x - \log_{a} y$ |

Sum of logarithms with common base | Show source$\log_{a} x + \log_{a} y = \log_{a} \left(x \cdot y \right)$ |

Difference of logarithms with common base | Show source$\log_{a} x - \log_{a} y = \log_{a} \frac{x}{y}$ |

Logarithm from power | Show source$\log_{a} x^n = n \log_{a} x$ |

Logarithm from n-th root | Show source$\log_{a} \left( \sqrt[n]{x} \right) = \frac{\log_{a} x}{n}$ |

Logarithm base conversion | Show source$\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$ |

# Decimal logarithm

Name | Formula |

Decimal logarithm from 1 | Show source$\log_{10} 10 = 0$ |

Decimal logarithm from 10 | Show source$\log_{10} 10 = 1$ |

Decimal logarithm from power of 10 | Show source$\log_{10} 10^n = n$ |

Decimal logarithm to natural conversion | Show source$\ln x = \frac{\log_{10} x}{\log_{10} e} \approx 2.3026 \cdot \log_{10} x$ |

Natural logarithm to decimal conversion | Show source$\log_{10} x = \frac{\ln x}{\ln 10} \approx 0.4343 \cdot \ln x$ |

# Natural logarithm

Name | Formula |

Natural logarithm from 1 | Show source$\ln e = 0$ |

Natural logarithm from e number | Show source$\ln e = 1$ |

Natural logarithm from power of e number | Show source$\ln e^n = n$ |

Decimal logarithm to natural conversion | Show source$\ln x = \frac{\log_{10} x}{\log_{10} e} \approx 2.3026 \cdot \log_{10} x$ |

Natural logarithm to decimal conversion | Show source$\log_{10} x = \frac{\ln x}{\ln 10} \approx 0.4343 \cdot \ln x$ |

# Some facts

- Calculation of
**logarithm**is an operation opposite to**exponentiation**. - To calculate the logarithm of base
**a**of the number**x**we ask the question:*"to what power we must raise the base to get x"*. - ⓘ Example: The decimal logarithm (of the basis 10) of the number one hundred (100) is two (2), because to obtain this number (100), we would have to raise the base of logarithm (10) to the
**second power**. Formally, we can write this down in the following way:

$\log_{10}100 = 2 \Leftrightarrow 10^2 = 100$ - Formally the logarithm is a
**two arguments operation**, where the first argument is the**logarithm base**, and the second one is the**number that we logarithm**. In general, we can write:

$\log_{a} x = b \Leftrightarrow a^b = x$where:

**a**- the logarithm base,

**x**- the number, which we logarithm,

**b**- the result of logarithm operation

- The logarithm is usually denoted by the symbol
**log**or**lg**. - Sometimes to simplify the expression we
**omit the base**of the logarithm. Then, by default, we assume that the basis is the**number ten**(10). - A special type of logarithm is a
**natural logarithm**. The basis of natural logarithm is so-called**number e**(sometimes also called the Euler's number):

$e = \sum\limits_{n = 0}^{\infty} \dfrac{1}{n!} = \dfrac{1}{1} + \dfrac{1}{1} + \dfrac{1}{1\cdot 2} + \dfrac{1}{1\cdot 2\cdot 3} + \cdots$ - In general, the logarithm is a
**inverse function for the power function $a^x$**. Thus, the natural logarithm is a**inverse to the exponential function $e^x$**. - The practical aspect of logarithm is that it allows to
**replace multiplication with addition**, which often simplifies calculations:

$\log_a(x \cdot y) = \log_a(x) + \log_a(y)$ - Logarithm is often used by
**engineers**and in**natural science**. This is because expressions that have an inherently**exponential character**(or more generally: power character), become**linear**after logarithm was applied. For this reason, many laws, e.g. in physics or chemistry have logarithm form.

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