Logarithm & ln Calculator

Calculate logarithmic values for any positive base, solve natural logs, and apply change-of-base rules.

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Enter the value and the base (such as 10 or e for natural logs) to solve the logarithm, and review the detailed calculations.

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A Guide to Understanding Logarithms

A logarithm is the mathematical inverse of an exponent power. In simple terms, a logarithm answers the question: "To what power must we raise a base number to get this value?" For example, since 10 raised to the power of 2 is 100, the logarithm of 100 with base 10 is exactly 2. Logarithms are essential for compressing massive ranges of physical values into readable, manageable scales. This makes them highly useful in scientific fields where numbers span many orders of magnitude.

Common Logarithmic Bases

While you can calculate logs for any positive base, three bases are used in most scientific applications. The common logarithm uses base 10 (often written simply as log). The natural logarithm uses Euler's constant e, which is approximately 2.71828 (written as ln). The binary logarithm uses base 2, which is heavily used in computing logic.

Since logarithms are the direct inverse of powers, they are closely linked to exponents, which you can calculate using our raising numbers to powers tool. They are also related to roots, which you can resolve using our calculating radical roots tool. For full formula support, check out our scientific calculator device tool.

Everyday Applications of Log Scales

Earthquake Magnitudes (Richter Scale): Each whole number increase on the scale represents a ten-fold increase in measured amplitude.
Sound Volumetrics (Decibels): Sound pressure is measured logarithmically because human ears perceive volume changes on a logarithmic scale.
Chemical Acidities (pH Scale): The acidity of water solutions measures hydrogen ion concentrations using a base-10 negative log scale.
Data Complexity: Computer scientists measure algorithm efficiency using binary logs, which you can analyze alongside our finding group averages tool or a simple arithmetic tool.

Logarithm Arithmetic Rules

Logarithms follow rules that turn multiplication into addition and division into subtraction. The product rule states: log(x × y) = log(x) + log(y). The quotient rule states: log(x / y) = log(x) - log(y). The power rule states: log(x^y) = y × log(x).

Additionally, you can solve logs with custom bases using the change-of-base formula: log_base_b(x) = ln(x) / ln(b). Our online solver automatically applies these rules, letting you evaluate complex equations quickly.

Example of Reversing a Power

Suppose you want to solve the logarithmic equation log base 2 of 32.

To solve this, we ask: "To what power must we raise 2 to get 32?" We multiply 2 by itself: 2 × 2 = 4, × 2 = 8, × 2 = 16, × 2 = 32. Since we multiplied 2 five times, 2 raised to the power of 5 is 32. Thus, the log base 2 of 32 is exactly 5. This basic example shows how logarithms decode exponent values.