Logarithm Calculator 2026 – Log, Ln & Any Base Solver

What Is a Logarithm Calculator?

The argument of a logarithm must be positive — log(0) and the log of any negative number are undefined. Calcrio’s free Logarithm Calculator evaluates log base 10, natural log (ln), logs of any base, applies the change-of-base formula, and solves log equations like log₂(x) = 5 step by step.

How to Use the Logarithm Calculator

Pick the mode that matches your problem, then enter your numbers. Every result shows its verification step so you can trust the answer.

1. Pick the mode that matches your problem.
2. Enter your numbers — invalid inputs are flagged as domain errors.
3. Select Calculate to generate the result.
4. Check the verification — every answer shows the b^y = x check.

What Is a Logarithm?

A logarithm is the inverse of an exponential: log_b(x) is the exponent you must put on the base b to get x. Written as a definition: log_b(x) = y ⟺ b^y = x (with b > 0, b ≠ 1, and x > 0).

Read log₂(32) = 5 as “log base 2 of 32 equals 5”, because 2 raised to the 5th power is 32. Two bases appear constantly, so they get their own names and buttons:

• Common logarithm — written log(x) with no base shown, always base 10. log(1000) = 3.
• Natural logarithm — written ln(x), always base e (≈ 2.71828). ln(e³) = 3.

Logarithms are everywhere in the real world: the pH scale of chemistry is −log₁₀ of hydrogen-ion concentration, the Richter scale for earthquakes is a base-10 logarithm of wave amplitude, sound is measured in decibels (a logarithmic unit), and compound interest, population growth, and radioactive decay all solve with logarithms.

Worked examples (exact numbers, verified)

• log₂(32) = 5, because 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.
• log₁₀(1000) = 3, because 10³ = 1000.
• ln(e³) = 3, by the inverse relationship ln(e^x) = x.
• log₅(125) = 3, because 5³ = 125.
• log₂(x) = 5 → x = 32: convert to exponential form x = 2⁵ = 32, then check log₂(32) = 5.

Powers and their logarithms (exact values)

Step-by-Step: Solving a Log Equation (and the Domain Trap)

The method is always: combine the logs into one, convert to exponential form, solve, then check the domain. Here is a classic exam question that punishes anyone who skips the last step:

Solve log₂(x) + log₂(x − 2) = 3

1. Combine with the product rule: log₂(x) + log₂(x − 2) = log₂(x(x − 2)). So log₂(x(x − 2)) = 3.
2. Convert to exponential form: x(x − 2) = 2³ = 8.
3. Solve the resulting quadratic: x² − 2x − 8 = 0 → (x − 4)(x + 2) = 0, so x = 4 or x = −2.
4. Check the domain — this is the trap. A logarithm’s argument must be positive. For log₂(x) we need x > 0; for log₂(x − 2) we need x > 2.
5. Reject the impostor: x = −2 makes both arguments negative (log₂(−2) is undefined), so it is extraneous. The only solution is x = 4.

Common Mistakes When Using Logarithms

The domain trap

Accepting x = −2 as a solution to log₂(x) + log₂(x − 2) = 3. Logarithms need positive arguments, so any “solution” that makes an argument zero or negative is extraneous. Check the domain before circling your answer.

Taking log(0) or log of a negative

log_b(x) is undefined for x ≤ 0 — there is no exponent that makes a positive base equal 0 or a negative number. If your calculator shows an error here, the input is wrong, not the calculator.

Confusing ln with log

ln is base e (≈ 2.71828); a bare “log” on most calculators and textbooks is base 10. ln(10) ≈ 2.303, but log₁₀(10) = 1 — mixing them up wrecks the answer.

Splitting log of a sum

log(x + y) is not log(x) + log(y). The sum rule works only on multiplication: log(xy) = log(x) + log(y). This is one of the most-tested errors in algebra exams.

Wrong change-of-base direction

log_b(x) = ln(x)/ln(b), not ln(b)/ln(x). For log₂(1000): 6.9078 ÷ 0.6931 ≈ 9.97 — flipping the division gives 0.1, which is obviously wrong.

Forgetting b ≠ 1

A logarithm base 1 does not exist (1^y is always 1, so it can never produce x). Any base of 1 is a domain error, just like a non-positive argument.

Benefits & Limitations

Benefits

• Five modes, one page: common log, natural log, any base, change of base, and log-equation solving — no switching tools mid-homework.
• Step-by-step, not just an answer: every mode shows the conversion, the computation, and the verification (b^y = x), so you can follow the work.
• Domain errors, not silent wrong answers: invalid inputs (log of 0, negative arguments, base 1) return a clear error message.
• Free and instant: no sign-up, works on phones for homework on the go.

Limitations

• Change-of-base results are decimal approximations (e.g. 9.9658 for log₂(1000)) because most logarithms are irrational; the calculator rounds for display.
• It evaluates and solves standard log forms — it is not a full computer algebra system and does not graph logarithmic functions.
• Like any calculator, it computes what you enter: it cannot fix a misread problem, so double-check which base your question uses (ln vs. log).

Quick Definitions

Related Math Tools

Logarithm Calculator Methodology

Every result on this page follows directly from the definition of the logarithm:

• Definition: log_b(x) = y ⟺ b^y = x, with b > 0, b ≠ 1, x > 0.
• Change of base: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). Verified: log₂(1000) = 6.907755/0.693147 ≈ 9.9658.
• Product rule: log_b(xy) = log_b(x) + log_b(y). Used to combine log₂(x) + log₂(x − 2) = log₂(x(x − 2)).
• Quotient rule: log_b(x/y) = log_b(x) − log_b(y).
• Power rule: log_b(xⁿ) = n · log_b(x).
• Inverse properties: b^(log_b(x)) = x and log_b(b^x) = x — the calculator uses these for its verification step.
• Domain check: before reporting any solution, each argument must satisfy x > 0 and the base must satisfy b > 0, b ≠ 1; values failing this are rejected as extraneous (the x = −2 case in the worked example).

Mathematical References

• Khan Academy — Logarithms (Algebra 2 unit)
• Paul's Online Math Notes (Lamar University) — Algebra: Logarithm Functions
• OpenStax — Algebra and Trigonometry