Log Calculator
Four modes: logarithm in any base (log₁₀, ln, log₂, custom), antilog, exponent aⁿ, and scientific notation. Results include verification and all three common bases simultaneously.
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How It Works
Select base 2, e, 10, 16, or type any custom base. The result always shows log₁₀, ln, and log₂ simultaneously — no need to recalculate. The verify line confirms the result: base^result = original number. Custom base uses the change-of-base formula: log_b(x) = ln(x) ÷ ln(b).
log_b(x) = ln(x) ÷ ln(b) | log₁₀(1000) = 3 | ln(e²) = 2Enter 1000, base 10 → log=3, ln=6.9078, log₂=9.9658Antilog is the exponentiation inverse of log. Enter the log value y and the base b — the result is b^y. Used to recover the original number after a log transformation. Example: if pH = 3, then [H⁺] = antilog(−3) = 10^−3 = 0.001 mol/L.
antilog_b(y) = b^y | antilog₁₀(3) = 10³ = 1000Enter y=3, base 10 → result 1000 | Verify: log₁₀(1000) = 3 ✓Computes base^exponent with three enhancements over a basic calculator: (1) BigInt for exact integer results up to 1000th power, (2) fractional exponents (2^0.5 = √2), (3) negative exponents with the 1/bⁿ explanation. Expanded multiplication steps show for small values.
aⁿ = a × a × ... × a | a^(−n) = 1/aⁿ | a^(1/n) = ⁿ√a2^10 = 1024 | 2^0.5 = 1.41421 | 10^−3 = 0.001Three sub-modes: (1) Standard → Scientific: enter any number, get coefficient × 10^exponent. (2) Scientific → Standard: enter in "1.23e5" or "1.23 × 10^5" format. (3) Arithmetic: perform +, −, ×, ÷ on two numbers already in scientific notation — result also in scientific form.
N = c × 10^e where 1 ≤ |c| < 100.000456 → 4.56 × 10^−4 | 3e8 × 2e3 = 6 × 10^11Three fundamental rules: Product rule: log_b(x×y) = log_b(x) + log_b(y). Quotient rule: log_b(x÷y) = log_b(x) − log_b(y). Power rule: log_b(xⁿ) = n × log_b(x). These rules let you break complex logs into simpler parts — the basis of how logarithm tables worked before calculators.
log(xy) = log(x) + log(y) | log(x/y) = log(x) − log(y) | log(xⁿ) = n·log(x)log(8) = log(2³) = 3·log(2) = 3×0.301 = 0.903 ✓Quick Reference
Common calculations — verify these instantly in the calculator above.
Log — base 10
log₁₀(1000)
3
Log — base e
ln(e²)
2
Log — base 2
log₂(1024)
10
Antilog
antilog₁₀(3)
1000
Antilog
antilog_e(1)
2.71828
Exponent
2^10
1024
Exponent
10^−3
0.001
Scientific
0.000456 → sci
4.56 × 10⁻⁴
Tips & Shortcuts
The Log tab always displays log₁₀, ln, and log₂ simultaneously — use these to cross-check or to switch between applications without re-entering the number.
For any base b logarithm not on the quick-pick buttons, type it directly in the base input field. Base 16 is useful for hex/bit calculations; base 2 for binary data sizes (log₂(1024) = 10 bits).
Fractional exponents in the Exponent tab are roots: 2^0.5 = √2, 8^(1/3) = ∛8 = 2, 16^0.25 = ⁴√16 = 2. Enter decimals directly.
Scientific notation input accepts both "1.23e5" (E-notation) and "1.23 × 10^5" — the calculator parses both formats. Copy-paste from a spreadsheet or paper both work.
The verify line on every result lets you confirm the answer without a second calculation: the Log tab shows base^result = original, and the Antilog tab shows log_b(result) = your input.
Common Mistakes
Using log₁₀ when a formula requires ln (or vice versa)
Growth, decay, and physics formulas almost always use ln (base e). Engineering and science scales (pH, dB, Richter) use log₁₀. Check the formula documentation — the symbol "log" without a subscript typically means base 10 in engineering and base e in pure math.
Entering a negative number or zero into the Log tab
Logarithms are only defined for positive real numbers. log(0) = −∞ and log(negative) is undefined in real numbers. The calculator will show an error — check that your input x > 0.
Confusing Antilog with the Log tab for the same operation
If you calculated log₁₀(x) = 3 and want to recover x, use the Antilog tab with y=3 and base 10 → result 1000. Using the Log tab again would give log₁₀(3) ≈ 0.477, which is a completely different calculation.
Entering scientific notation in wrong format (e.g. "1.23 * 10^5")
Use either 1.23e5 or 1.23 × 10^5. The × symbol must be × (multiplication symbol) or the calculator will reject the format. Copy-paste from standard math notation or use the E-notation shorthand.
Expecting log₂(1024) = 100 because 1024 looks like a round number
log₂(1024) = 10 because 2^10 = 1024. Logarithms grow very slowly — log₂(1,000,000) ≈ 19.93. A result above 50 for base 10 or base 2 usually means a data entry error.
Frequently Asked Questions
log (log₁₀) is base-10: log(1000) = 3 because 10³ = 1000. ln is base-e: ln(e) = 1, ln(e²) = 2. Use log for pH (−log[H⁺]), decibels (10·log ratio), and the Richter scale. Use ln for continuous compound interest (A = Peʳᵗ), radioactive decay, and population growth models. The calculator shows both simultaneously for every input.
Select the Log/Ln tab, enter your number, then type any base in the base input field — it accepts any positive number except 1. The calculator uses the change-of-base formula internally: log_b(x) = ln(x) ÷ ln(b). Example: log₅(125) — enter 125, type base 5, press Calculate → result 3. Verify: 5³ = 125 ✓
Antilog is the inverse of a logarithm. antilog_b(y) = b^y. If log₁₀(x) = 3, then antilog₁₀(3) = 10³ = 1000. In the Antilog tab: enter the log value (y), select or type the base, press Calculate. The result is the original number x, with a verify line showing log_b(result) = your input.
Two accepted formats: 1.23 × 10^5 or 1.23e5. For arithmetic between two scientific notation numbers, use the arithmetic sub-mode — enter each number separately and select +, −, ×, or ÷. Result shows both scientific and standard forms with the coefficient and exponent separated.
The Exponent tab computes aⁿ directly — it raises a base to any power. Fractional exponents: 2^0.5 = 1.41421 (square root of 2). Negative exponents: 2^−3 = 0.125 = 1/8. Large integer powers: 2^64 computed exactly using BigInt. Quick-pick buttons (2², 3², 10³, e², 2^−1, 2^0.5) let you test common cases instantly.
Because the three most common bases serve different fields: base 10 for engineering and science, base e for calculus and natural processes, base 2 for computer science. Showing all three simultaneously means one calculation covers every common use case. log(x) = ln(x) ÷ ln(10) and log₂(x) = ln(x) ÷ ln(2) — all derived from the same ln(x) computation.
Solving for y (the log): use the Log/Ln tab with x and base b. Solving for x (antilog): use the Antilog tab — enter y and base b, result = b^y. Solving for b: b = x^(1/y) — use the Exponent tab with base x and exponent 1/y. Example: log_b(32) = 5, so b = 32^(1/5) = 2. Enter 32 as base and 0.2 as exponent → result 2.
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