Log Calculator

Our Log Calculator lets you compute logarithms quickly and accurately for any positive number and any valid base, including base 10 (common log), base e (natural log), base 2, and a custom base of your choice.

What is a logarithm?

A logarithm answers the question: to what exponent must we raise a given base to obtain a specified number? If b is the base and x is the number, then y = log_b(x) means b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000. Logarithms are fundamental in mathematics, science, engineering, computer science, and data analysis because they transform multiplicative relationships into additive ones and compress large ranges of values into manageable scales.

How to use the Log Calculator

1. Enter the positive number you want the logarithm of (the argument).
2. Select the base: 10, e, 2, or choose Custom to enter any valid base.
3. Optionally, set the decimal precision to control the number of digits after the decimal point.
4. Click Calculate to get your result instantly.

Remember the domain restrictions: the argument must be greater than zero. The base must be greater than zero and cannot be equal to one.

Formula and change-of-base

The calculator uses the change-of-base formula for consistency and precision across all bases:

log_b(x) = ln(x) / ln(b)

Here ln denotes the natural logarithm (logarithm base e). This approach ensures a single, reliable method regardless of whether you choose base 10, base 2, e, or a custom base.

Why change-of-base matters

While some programming languages offer a direct log function for common bases, the change-of-base formula provides a universal way to compute logarithms using the natural log. It also makes it easy to show steps: compute ln(x), compute ln(b), then divide ln(x) by ln(b).

Common use cases

• Scientific and engineering: pH, decibels, Richter scale, and radioactive decay rely on logarithmic relationships.
• Computer science: Algorithmic complexity (e.g., O(log n)), binary trees, and data structures often involve log base 2.
• Data analysis: Log transforms stabilize variance, linearize exponential growth, and aid in regression modeling.
• Finance: Continuously compounded returns and growth models naturally use the natural log.

Tips for accurate results

• Ensure the argument is strictly positive; logarithms are undefined for zero or negative inputs.
• Choose a base greater than 0 and not equal to 1; otherwise, the logarithm is undefined.
• Adjust decimal places to balance readability and precision. For most practical uses, 4–8 decimal places are sufficient.
• Use base e for natural logs (ln), base 10 for common logs, and base 2 for binary computations.

Examples

log₁₀(1000) = 3, because 10³ = 1000. ln(e) = 1, because e¹ = e. log₂(256) = 8, because 2⁸ = 256. For a custom example, log₃(81) = 4, since 3⁴ = 81.

Troubleshooting

• Got an error? Check that your number is greater than 0 and your base is greater than 0 and not equal to 1.
• Unexpected precision? Increase the decimal places setting to see more digits of the result.
• Custom base not applied? Ensure you selected the Custom option and entered a valid custom base.

Why use this Log Calculator?

It’s fast, supports multiple bases, includes precision control, and can display the change-of-base steps for learning and verification. Whether you’re a student, researcher, engineer, or developer, this tool provides clear, dependable log computations.