Calculator Tools
Logarithm Calculator
Compute log base 10, ln, log base 2, or log with any base. Antilog, change of base, and solve log_b(x) = y for any unknown. Step-by-step.
Logarithm calculator
Choose a mode
All four modes run locally with native browser math; nothing is uploaded.
Compute log_b(x) for any base. Presets for log₁₀, ln, and log₂.
Must be positive. Accepts decimals, simple fractions like 3/4, and the symbolic constants e and pi.
Must be positive and not equal to 1. Type e for natural log.
Result
3
log(1000) = 3
log₁₀(1000)
3
ln(1000)
6.907755279
log₂(1000)
9.9657842847
Step-by-step
Each line shows the substitution used to reach the next result.
- Read the logarithm as: log(1000) is the exponent y such that 10^y = 1000.
- Use the change-of-base identity: log(1000) = ln(1000) / ln(10) = 6.907755279 / 2.302585093.
- Divide to get the result: log(1000) = 3.
- Verify by exponentiating: 10^3 = 1000 (should match the original argument).
What a logarithm is
- log_b(x) = y means b^y = x. The log answers the question: what exponent do I raise the base to in order to reach this argument?
- The argument x must be strictly positive. The base b must be positive and not equal to 1, because base 1 has no inverse and negative bases would force complex results.
- Three special bases get their own names. Base 10 is the common log (often written log without a subscript). Base e is the natural log (written ln). Base 2 is the binary log (often written log₂ or lg).
Useful identities
- log_b(x y) = log_b(x) + log_b(y) turns multiplication into addition.
- log_b(x / y) = log_b(x) - log_b(y) turns division into subtraction.
- log_b(x^p) = p log_b(x) brings exponents down as a coefficient.
- log_b(x) = log_k(x) / log_k(b) is the change-of-base formula, useful when your calculator only has a few bases.
- log_b(1) = 0 and log_b(b) = 1 for every valid base.
How to use
- Pick a mode: Logarithm to evaluate log_b(x); Antilog to compute b^y; Change of base to rewrite log_b(x) using bases 10, e, and 2; or Solve equation to solve log_b(x) = y for the missing variable.
- In Logarithm mode, type the argument x, then type the base b or tap one of the log₁₀, ln, or log₂ presets. The result panel shows the headline log_b(x) plus the three standard bases at once.
- In Antilog mode, type the base b and the log value y. The result panel returns x = b^y and verifies it by taking the log back to confirm.
- In Change of base mode, type the argument x and the original base b. The calculator evaluates log_b(x) three different ways using the change-of-base formula. Optionally add a custom intermediate base k for a fourth route.
- In Solve mode, choose which value to solve for (y, x, or b). Type the other two values. The calculator returns the missing variable and shows the algebra used to isolate it.
- Read the Step-by-step block to see each substitution and the verification. Use Copy result or Copy steps to paste into homework, notes, or a write-up.
- Click Load example to populate a familiar problem (log₁₀(1000) = 3), or click Clear to start over.
About this tool
Logarithm Calculator handles the four logarithm problems that come up most often in homework, scientific work, and engineering: evaluating log_b(x) for any base, taking an antilogarithm to recover the argument from a known log value, rewriting a log in a more convenient base using the change-of-base formula, and solving log_b(x) = y for whichever of the three values is missing. The Logarithm mode reports the result for an arbitrary base and also shows the three standard bases side by side, so you can read log₁₀, ln (base e), and log₂ in a single glance without re-entering the argument. The Antilog mode raises a chosen base to a chosen exponent, which is the inverse direction students often have to reach for at the end of a problem. The Change of Base mode evaluates log_b(x) along three routes (base 10, base e, base 2) and an optional fourth user-supplied base, so the textbook identity log_b(x) = log_k(x) / log_k(b) is visible as numbers, not just as symbols. The Solve mode handles the two most common homework prompts (find x given b and y; find y given b and x) plus the harder case of solving for the base itself, including the cases where the equation has no unique base. Inputs accept integers, decimals, comma-decimals used in European locales, simple fractions like 3/4, and the symbolic constants e and pi so you can type ln(e) by selecting the natural log and entering 'e' as the argument. Every result is shown with a step-by-step derivation that mirrors the work you would hand in, including the change-of-base substitution and the verification step where the answer is exponentiated back to the original argument. Computation runs entirely in your browser using native Math primitives; the numbers you type here never leave your device.
Free to use. Works in your browser. No signup, no login.
Related tools
You may also like
Quadratic Equation Solver
Roots, discriminant, vertex, factored form, and step-by-step working for any quadratic.
Open tool
CalculatorSlope Calculator
Slope, line equation, midpoint, distance, and angle from two points.
Open tool
ConverterScientific Notation Converter
Convert between scientific, E, engineering, and decimal forms with a precision slider.
Open tool
CalculatorFraction Calculator
Add, subtract, multiply, and divide fractions with simplified results and step-by-step working.
Open tool
CalculatorPermutation and Combination Calculator
nPr, nCr, factorial, n^r, and multiset coefficient with exact step-by-step working.
Open tool