Log Calculator
Enter a base and a number to get the logarithm of that number in that base. The result also shows the natural log, ln(x), and the common log, log₁₀(x), of the same number. The base can be any positive number other than 1, such as 10, 2 for a binary log, or 0.5.
How to use the Log Calculator
- Enter your numbers in the Log Calculator above.
- Press the Calculate button to see the result.
- To try other numbers, change any value and press the button again.
The logarithm definition
log_b(x) = y ⟺ b^y = x · · · Change of base: log_b(x) = ln(x) / ln(b)
The logarithm of x with base b is the exponent to which b must be raised to produce x. The calculator uses the change-of-base rule, log_b(x) = ln(x) ÷ ln(b), and rounds every result to 6 decimal places. A result closer to 0 than 0.000001 shows in scientific notation with 6 significant digits instead, so log₁₀(1.0000001) shows 4.34294 × 10^-8.
Worked example
log₁₀(1000) = 3 (because 10³ = 1000). log₂(64) = 6 (because 2⁶ = 64). ln(7.389) ≈ 2 (because e² ≈ 7.389), and the calculator shows 1.999992. log₅(125) = 3 (because 5³ = 125).
Frequently asked questions
What's the difference between log and ln?
log usually means log₁₀, the common log with base 10. ln means log_e, the natural log with base e ≈ 2.718. Which one a plain "log" means depends on the context. The calculator shows both next to the log in the base you enter.
Why is log of a negative number undefined?
No real exponent makes a positive base produce a negative result, or zero. Complex logarithms exist for negative numbers, but they have an imaginary part. The calculator asks for a number greater than 0.
What is log of 1?
Always 0, whatever the base, because any base raised to 0 equals 1.