Logarithm Calculator
Calculate the logarithm of a number to any base, including common log (base 10) and natural log (base e).
Your results
Calculation breakdown
- Logarithm
- log_base(x) = ln(x) ÷ ln(base)
Worked example
log base 10 of 100 is 2, since 10 squared equals 100.
Assumptions
- Logarithms of zero or negative numbers are undefined and return an error.
- The base must be a positive number other than 1.
How this calculator works
This calculator finds the logarithm of a positive number to a base you choose, including base 10 and base e (natural log). It uses the change-of-base formula to support any positive base other than 1.
What a logarithm means
A logarithm answers the question: what exponent must the base be raised to in order to produce the number? If 10² = 100, then log base 10 of 100 is 2.
Common logarithms use base 10, while natural logarithms use the constant e. The change-of-base formula lets the calculator evaluate other valid bases using logarithms available on a standard calculator.
- log base b of x = y means bʸ = x
- Change of base: log_b(x) = ln(x) ÷ ln(b)
- The argument x must be greater than zero for a real result
Valid bases and rounding
For real-number logarithms, the base must be positive and cannot equal 1. A zero or negative argument does not produce a real logarithm, so check the inputs if the result is undefined.
Displayed decimals are rounded. Keep extra precision during intermediate calculations if the logarithm will be used in a later formula.
Frequently asked questions
What is a logarithm?
A logarithm answers the question: to what power must the base be raised to produce this number? For example, log base 10 of 100 is 2, since 10 squared is 100.
Can I find a logarithm of a negative number?
No, logarithms of zero or negative numbers aren't defined for real numbers, so the calculator shows an error.
What is natural log used for?
Natural log (base e) commonly appears in continuous growth and decay problems, such as compound interest calculated continuously.
Editorial review and responsibility
Reviewed by the CalcAussie editorial team on 21 September 2026. Review scope: Method, Australian terminology, material assumptions, source links and user-facing limitations.
Calculator results remain estimates based on the figures entered. Check material decisions against current official information and report a suspected error so it can be reproduced and corrected.