Logarithm Calculator
Compute logarithms in any base, like log₂(8). One-tap bases for 2, 10 and e, with the change-of-base working (ln x ÷ ln b) and a check that bʸ returns x. Common and natural logarithm notes included.
Common bases
e.g. 8, 1000, e, 0.5
Enter a base and an argument to see the logarithm.
How to use
- 1
Enter the base
Type the base b, or use the one-tap buttons for 2, 10 and e.
- 2
Enter the argument
Type the argument x, for example 8, 1000 or 0.5. The letter e works too.
- 3
Read value and reasons
Alongside the value, see the substituted change-of-base formula and the check that bʸ returns x.
Features
- Any-base logarithms with one-tap bases for 2, 10 and e (e also works as the argument)
- Exact integer display when the value is whole; decimals to six places otherwise
- Change-of-base substitution steps plus a bʸ = x verification
- Common and natural logarithm notes included; inputs are never transmitted
Use cases
Check math homework
Verify hand-worked logarithm problems together with the checking steps.
Estimate bits and digits
Use log₂ to count how many powers of two fit, or gauge digit counts.
Support science formulas
Pull values that feed into pH, decibels and other logarithmic quantities.
Details
A logarithm log_b(x) answers the question "to what power must b be raised to get x": since 2³ = 8, log₂8 = 3. Enter a base and an argument and the tool computes the value; when the result is an exact integer (log₂8 = 3, log₃81 = 4), it is shown without floating-point noise.
A calculator’s log key only covers base 10 (the common logarithm) and base e (the natural logarithm, ln), so bases like 2 need the change-of-base formula log_b(x) = ln x ÷ ln b. The tool substitutes your numbers into this formula step by step, so you can see why the value comes out as it does. With base 10 you also get digit estimates such as ⌊log₁₀x⌋ + 1.
The conditions are x > 0 for the argument, and b > 0 with b ≠ 1 for the base: logs of zero or negative numbers do not exist, and base 1 gives no defined value. A verification block computes bʸ to confirm it returns the argument, including how well the rounded decimal matches. Inputs are never transmitted; everything is computed locally in your browser.
FAQ
What is a logarithm?
It is the exponent that answers "what power of the base gives the argument": 2³ = 8 means log₂8 = 3. Logarithms restate exponents and turn multiplication into addition (log ab = log a + log b), which greatly simplifies calculation.
How is this different from a calculator’s log key?
A calculator’s log is base 10 (common logarithm) and ln is base e (natural logarithm) only. Other bases need the change-of-base formula log_b(x) = ln x ÷ ln b, and this tool shows that substitution step by step.
What happens if the argument is 0 or negative?
The tool reports an error. b^y is always positive, so by definition the argument of a logarithm must be positive (x > 0).
Why can’t the base be 1?
One raised to any power stays 1, so 1^y = x either has no solution or no unique one. By definition the base must be positive and not equal to 1.
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All processing happens in your browser. Your files are never uploaded.
Verified: Known values (log₂8, log₃81, log₅(1/125), ln e), argument and base error handling, the change-of-base display and the English page are covered by browser tests
Did you know?
Counting digits in base 10 is the same as taking log₁₀. Since log₁₀2000 falls between 3 and 4, 2000 has four digits — the relation "digits = ⌊log₁₀x⌋ + 1".