Logarithm calculator
Common, natural and any-base logarithms with a check.
Fill in the fields and the result will appear here automatically.
Finds the exponent to which the base must be raised to give the number. All three modes use one formula, ln x divided by ln b, and the result comes with a check by exponentiation.
How it works
Formula and logic
log_b(x) = ln x ÷ ln b; the common and natural modes only fix the base. The number is positive and the base positive and not 1; bases between 0 and 1 are valid. Ordinary decimal output is rounded to six places; very small numbers use scientific notation and the check is approximate.
Example
For x = 1024 and base 2 the result is 10. For x = 4 and base 0.5 the result is −2, since 0.5⁻² = 4. For x = 1 the logarithm is zero for every valid base; x = 0 is invalid.
Fields and units
- Logarithm type — list option
- Number — unitless
- Base — unitless
How to use
- — Choose common, natural or custom-base logarithm.
- — Enter a positive number; for a custom base enter a positive base other than one.
- — Read the exponent and approximate check; fixed-base modes ignore the base field.
Method and limitations
- Calculation method
- Formula and logic
- Limitation
- Uses finite real numbers. Complex logarithms are not supported; the result and check are rounded.
FAQ
Why must the number be positive?
No power of a positive base ever gives zero or a negative number, so the logarithm has no value there.
Why can the base not be one?
One raised to any power is still one, so the equation has no single answer.
What is e?
The base of natural logarithms, about 2.71828. It appears wherever growth is continuous.
What is the check line for?
It raises the base to the rounded exponent. Floating-point arithmetic and rounding make this an approximate check, not a proof of exact equality. Integer powers such as 2¹⁰ = 1024 usually match exactly.