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Evaluate log10x
Evaluate log2x
Evaluate ln(x)
Evaluate n-th power
Evaluate n-th root
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Logarithm to base n calculator

- By Dr. Minas E. Lemonis, PhD - Updated: March 3, 2019
Home > Evaluation > n-th logarithm

This tool evaluates the base n logarithm of a number: \log_{n}{x} . Enter the base n and the argument x below.

n =
x =
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Result:
\log_{n}{x} =
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Table of Contents
-  Calculator
-  Definitions
-  General
-  Common bases
-  Properties
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Definitions

General

The logarithm of a number x to a base n, is a power p so that np= x. In modern notation logarithm is written as \log_{n}{x} . Any positive real number has a logarithm to a positive real base (except base 1). For any valid base, logarithm of 1 is 0, while the logarithm of 0 approaches negative infinity. Changing between bases can be done with the formula:

\log_{n} x = \frac{\log_{m}{x}}{\log_{m}{n}}

Common bases

In practice, some logarithmic bases find more frequent use than others. Such common bases are 10 (decimal logarithm, \log_{10}{x} or \lg{x} ), 2 (binary logarithm, \log_{2}{x} ) and the mathematical constant e≈2.718, (natural logarithm, \ln{x} )

Properties

The derivative of the logarithm function is:

\left(\log_{n} x\right)' = \frac{1}{x}\frac{1}{\ln{n}}

The integral of the decimal logarithm function is given by:

\int \log_{n} x\, \mathrm{d}x = \frac{x\ln{x}-x}{\ln{n}} +C

Identities:

\begin{split} & \log_{n} \left(x y\right) & = \log_{n}{x} + \log_{n}{y} \\ \\ & \log_{n} \left(\frac{x}{y}\right) & = \log_{n}{x} - \log_{n}{y} \\ \\ & \log_{n} \left(x^p\right) & = p\log_{n}{x} \\ \\ & \log_{n} \left(\sqrt[p]{x}\right) & = \frac{\log_{n}{x}}{p} \\ \\ & \log_{n} {n} & = 1 \\ \\ & \log_{n} {n^x} & = x \\ \\ \end{split}

See also
Evaluate log10x
Evaluate log2x
Evaluate ln(x)
Evaluate n-th power
Evaluate n-th root
All evaluation tools