Logarithm Calculator
Compute logarithms, natural logs, and change-of-base results.
x must be > 0
b must be > 0 and b ≠ 1
Result
Solve a simple log equation
Equation format: log_b(x) = y
We use x = bʸ.
Logarithm formulas used
General definition:
\(\log_b(x) = y \iff b^y = x\)
Change of base:
\(\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}\)
Common log (base 10): \(\log_{10}(x)\)
Natural log (base e): \(\ln(x)\)
Common logarithm values
| x | log₁₀(x) | ln(x) | Notes |
|---|---|---|---|
| 1 | 0 | 0 | log of 1 is always 0 |
| 2 | 0.3010 | 0.6931 | - |
| 10 | 1 | 2.3026 | base 10 |
| 100 | 2 | 4.6052 | 10² |
| 0.1 | -1 | -2.3026 | less than 1 → negative log |
| e ≈ 2.71828 | 0.4343 | 1 | by definition |
How to use this logarithm calculator
- Enter the number x you want to take the log of.
- Enter the base b, or click one of the quick buttons (10, e, 2).
- The result updates instantly using the change-of-base formula.
- If you need to solve log_b(x) = y, use the blue box to compute x = bʸ.
If the calculator shows an error, check that x > 0 and b > 0 and b ≠ 1.
FAQs
Can I use this for logarithmic equations?
Yes, for basic equations of the form log_b(x) = y. For more complex ones (like log(x) + log(x-1) = 2), you might need a CAS or symbolic solver.
Why is the result negative?
Logs of numbers between 0 and 1 are negative. For example, log₁₀(0.01) = -2.
What base should I use in engineering?
Base 10 (common log) is widely used in engineering and signal processing. Natural log (base e) is common in calculus, physics, and continuous growth models.
Formula (LaTeX) + variables + units
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General definition: \(\log_b(x) = y \iff b^y = x\) Change of base: \(\log_b(x) = \frac{\ln(x)}{\ln(b)} = \frac{\log_{10}(x)}{\log_{10}(b)}\) Common log (base 10): \(\log_{10}(x)\) Natural log (base e): \(\ln(x)\)
- No variables provided in audit spec.
- NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures - FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/
Last code update: 2026-01-19
- Initial audit spec draft generated from HTML extraction (review required).
- Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
- Confirm sources are authoritative and relevant to the calculator methodology.