Logarithm Calculator
Compute log_b(x) for any positive base with the change-of-base formula log_b(x) = ln(x) ÷ ln(b) — the full working is shown.
Reviewed by the OmniCalc teamMethod verified 2026-07-01
3
Result log_10(1,000) = 3Show steps
- Change-of-base formula: log_b(x) = ln(x) ÷ ln(b).
- Substitute x = 1,000 and b = 10: log_10(1,000) = ln(1,000) ÷ ln(10).
- = 6.907755 ÷ 2.302585 = 3.
How to use the logarithm calculator
- 1Enter the number (x) — the value you want the logarithm of. It must be greater than zero.
- 2Enter the base (b) — any positive number except 1. Leave it at 10 for the common log, or use e for the natural log.
- 3Read logb(x), computed instantly by the change-of-base formula.
- 4Open Show steps under the result to follow the exact arithmetic.
A logarithm is an exponent in reverse
Where a power asks “what is bⁿ?”, a logarithm runs it backwards: logb(x) asks “to what power must I raise b to get x?” So log₂(8) = 3 because 2³ = 8. That inverse link is why logarithms and exponents are always taught together.
Frequently asked questions
What is a logarithm?
A logarithm answers the question “what power do I raise the base to in order to get this number?” Writing log_b(x) = y means bʸ = x, so log₁₀(1000) = 3 because 10³ = 1000.
What is the change-of-base formula?
Any logarithm can be rewritten with natural logs: log_b(x) = ln(x) ÷ ln(b). This calculator uses it so you can take a log to any positive base, not just 10 or e.
What base should I use?
Base 10, the common log, is the default and is handy for orders of magnitude. Base e ≈ 2.71828 gives the natural log (ln), and base 2 is common in computing. Any positive base other than 1 works.
Why do I get no result for some inputs?
A logarithm is only defined when the number is positive (x > 0) and the base is positive and not 1 (b > 0, b ≠ 1). Zero, negative numbers, or a base of 1 have no logarithm, so the calculator shows a dash.
What is the difference between ln and log?
ln is the natural log, base e; “log” on its own usually means the common log, base 10. Both relate to your custom-base result through the change-of-base formula log_b(x) = ln(x) ÷ ln(b).