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.
Inputs are not collected. Usage measurement requires your consent.
Common bases
e.g. 8, 1000, e, 0.5
Enter a base and an argument to see the logarithm.
Done
Logarithm Calculator
Next time, open this tool from My Tools. Only the tool list is saved in this browser, not your inputs or results.
⌥DPress Option+D anytime to toggle My Tools instantly
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)
- Floating-point results: up to six decimal places, or scientific notation for small nonzero values
- 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; results use floating-point arithmetic. Known values such as log₂8 = 3 are displayed as integers, but rounding is not a proof of exactness.
A calculator’s log key usually means base 10 and ln means base e; other bases can use 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. For positive integers x, digits = ⌊log₁₀x⌋ + 1; do not decide a boundary from a rounded display.
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.
Related guide
GuideHow to Use the Laws of Exponents: Product, Quotient, Power and More
Compare exponent laws, negative bases and reciprocals. Distinguish −2⁴ from (−2)⁴, understand the 0⁰ convention, fractional-exponent limits, scientific notation for 10⁻⁷ and exact versus approximate calculation.
GuideHalf-Life Calculation: Formula, Examples, and Common Mistakes
Calculate remaining amount, elapsed time and half-life with N = N₀(1/2)^(t/T). Worked examples with carbon-14 and cobalt-60, log₂ back-calculation, and common mistakes such as using e^(−t/T) or confusing half-life with mean life.
GuideHow to Calculate Logarithms: Rules, Change of Base and Examples
Calculate logarithms from the definition, including bases below one. Worked change-of-base examples, log vs ln, domain checks, digit-count boundaries, rounded outputs, a diagram and FAQs.
Related Tools
Exponent Calculator
Calculate powers from a base and exponent. Safe integer bases with nonnegative integer exponents use BigInt; negative exponents and decimals are approximate. Inspect signs, scientific notation and the 0⁰ convention.
Half-Life Calculator
Compute the remaining amount after any elapsed time with N = N₀ × (1/2)^(t/T). Includes carbon-14 dating (5730 years) and drug-metabolism examples, plus the remaining percentage, half-lives elapsed and the decay constant λ.
Trigonometry Calculator
Enter an angle to get sin, cos and tan at once. Special angles like 30°, 45° and 60° are shown as exact values (1/2, √3/2), with degree and radian support, pi expressions such as pi/6, plus the quadrant signs and the reference-angle working.
Fraction Calculator
Add, subtract, multiply and divide fractions with exact results. See the reduced answer plus mixed-number and decimal forms, with the working shown.
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".