Logarithm calculator
Choose base 10, base e, or a custom base, then enter a positive number to find its logarithm.
Logarithm (logᵦ x)
Find the exponent that produces a positive number.
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Result
Enter the values, then select Calculate.
How to use the logarithm calculator
A logarithm finds an exponent: logᵦ(x) = y means bʸ = x. Choose the familiar base 10, natural base e, or enter another valid base. Calculations stay in your browser.
Logarithms: common log, natural log, and a custom base
A logarithm answers an exponent question. If bʸ = x, then logᵦ(x) = y. Common logarithm uses base 10, so log₁₀(1,000) = 3. Natural logarithm uses Euler’s number e, so ln(e) = 1. Custom base supports cases such as log₂(64) = 6.
The number x must be greater than zero. The base must also be greater than zero and cannot equal 1. Bases between 0 and 1 are valid and produce decreasing logarithmic functions. For example, log₀·₅(4) = −2 because 0.5⁻² = 4. The calculator uses the change-of-base formula ln(x) ÷ ln(b) for custom bases.
Common, natural, and custom-base examples
Common log uses base 10, so log₁₀(1,000) = 3. Natural log uses e, so ln(e) = 1. For a custom base, log₂(64) = 6 because 2⁶ = 64.
A base between 0 and 1 is valid: log₀·₅(4) = −2 because 0.5⁻² = 4. The number must always be positive.
Domain rules and precision
- The logarithm argument must be greater than zero.
- A custom base must be greater than zero and cannot equal 1.
- Custom-base results use the change-of-base formula ln(x) ÷ ln(b).
- Non-integer answers are normally approximate floating-point values.
- Very large or small finite answers may use scientific notation.
Frequently asked questions
What is the difference between log and ln?
Common log uses base 10, while ln uses Euler’s number e as its base.
Can I use a custom logarithm base?
Yes. Choose Custom base and enter a positive base other than 1.
Why must the number be positive?
No real exponent of a positive logarithm base produces zero or a negative number.
Why can the base be below 1 but not equal 1?
A positive base below 1 still changes one-to-one with the exponent. Base 1 always stays 1, so it cannot define a unique logarithm.
How can I check a logarithm result?
Raise the selected base to the returned exponent. It should reproduce the entered number within normal rounding precision.