Logarithm with Base Calculator

Formula

Logarithm = ln(Positive number) ÷ ln(Logarithm base)

Worked example

With Positive number: 100, Logarithm base: 10, logarithm is 2.

Calculation method

The rule is Logarithm = ln(Positive number) ÷ ln(Logarithm base).

The logarithm is the exponent required to raise the chosen base to the given positive number. The base must be positive and cannot equal one.

For this logarithm with base example, changing positive number from 100 to 101 while keeping the other inputs fixed changes logarithm from 2 to 2.004321.

Reading logarithm

The output is an exponent. A positive output means the number exceeds one for a base greater than one; zero corresponds to an input of one. In the worked values on this page, positive number 100, logarithm base 10 produce logarithm 2.

Practical use

Use logarithms to solve exponential growth relationships or compare quantities on a logarithmic scale. The specific task here is to find logarithm from positive number, logarithm base.

Limits and common mistakes

The argument must be positive. A chosen base must be positive and different from one; logarithms with these invalid inputs have no real result. Verify positive number before interpreting logarithm; a value measured on another basis would answer a different question.

Why the relationship works

A logarithm is the inverse question to exponentiation: which exponent produces the positive input from the selected base? Natural logarithm uses base e; a chosen-base logarithm divides natural logs to change the base. The output is an exponent and need not be a whole number. The logarithm with base rule uses positive number, logarithm base in Logarithm = ln(Positive number) ÷ ln(Logarithm base); read each symbol as the field named on this page.

Interpreting another set of values

For logarithm with base, the fields labeled positive number, logarithm base are the known quantities. The calculator solves for logarithm using Logarithm = ln(Positive number) ÷ ln(Logarithm base). The result may be an exact integer, a rounded decimal, an exponent, or a principal root. Preserve the original fraction or expression when a rounded display would lose precision needed later. When comparing two calculations, keep their units and input definitions identical.

For a second logarithm with base check, keep the other default inputs fixed and change positive number from 100 to 102. Logarithm changes from 2 to 2.0086. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

The logarithm argument must be positive. A chosen base must be positive and different from one. The logarithm of one is zero for every valid base, which gives a convenient check before considering rounded decimal output. The reported logarithm is 2 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.