About this Log Calculator
Logarithms are the inverse operation to exponentiation. They answer the question: 'To what power must we raise the base to get this number?' They are widely used in science to measure scales like pH, the Richter scale (earthquakes), and decibels (sound).
Who this calculator is for
- Students learning inverse operations, exponents, or logarithm rules.
- Scientists and engineers working with scales that grow by powers.
- Anyone checking log, ln, or custom-base results quickly.
When to use it
- Use this calculator to solve for the exponent in a power equation.
- Use it for common log, natural log, or custom-base log problems.
- Use it when you need to work with exponential growth or scale-based measurements.
How to use this calculator
- Enter the Value (x) you want to find the log of.
- Enter the Base (Common default is 10).
- The result shows both the Log in that base and the Natural Log (ln).
Key assumptions
- The value being logged must be greater than 0.
- The base must be greater than 0 and cannot equal 1.
- Results are returned in real numbers only.
Accuracy and testing
Review formula sources, default test cases, assumptions, unit handling, rounding rules, and changelog for this calculator.
How this calculator works
The logarithm $\log_b(x) = y$ is equivalent to $b^y = x$.
* Common Log: Base 10 (written as $\log x$).
* Natural Log: Base $e$ (written as $\ln x$).
* Binary Log: Base 2 (used in computer science).
Inputs explained
Value (x)
The number you are analyzing. Must be greater than 0.
Base
The base of the logarithm. Common bases are 10, 2, and $e$ (approx 2.718). Must be positive and not equal to 1.
Understanding your results
The result is the exponent. For example, $\log_{10}(100) = 2$ because $10^2 = 100$.
Example scenarios
Base 10
Log(1000) base 10 = 3 (since $10^3 = 1000$).
Base 2
Log(16) base 2 = 4 (since $2^4 = 16$).
Natural Log
Ln(7.389) ≈ 2 (since $e^2 \approx 7.389$).
Key Features
- Custom Base Support
- Calculates ln simultaneously
- Handles decimals
Tips & Best Practices
- Logarithms turn multiplication into addition: $\log(ab) = \log(a) + \log(b)$.
- They turn exponents into multiplication: $\log(a^n) = n \cdot \log(a)$.
- You cannot take the log of a negative number or zero in the real number system.
Glossary of Terms
- Natural Logarithm (ln)
- A logarithm with base $e$ (Euler's number, ~2.71828).
- Common Logarithm
- A logarithm with base 10.
- Exponential Growth
- Growth whose rate becomes ever more rapid in proportion to the growing total number or size. The inverse of logarithmic growth.