Logarithm Calculator

A logarithm is the inverse of exponentiation, and this calculator evaluates it for any base you need. It answers a single question: "To what power must the base be raised to produce this number?" Written formally, if b^y = x then log_b(x) = y. Here b is the base (a positive number not equal to 1), x is the argument (which must be strictly positive), and y is the result the calculator returns. Because raising any positive base to a real power can never give zero or a negative number, the logarithm is undefined for x ≤ 0, which is why the tool shows a warning instead of a value when you enter a non-positive argument.

This page is built for students working through algebra, pre-calculus, or calculus homework; for engineers and scientists who deal with decibels, pH, or signal processing; for computer-science learners who need base-2 logs for complexity analysis; and for anyone who simply wants to check an answer quickly. It computes the common log, the natural log, the binary log, and a custom-base log side by side, and it includes an anti-logarithm panel so you can move back and forth between a number and its exponent without reaching for a separate tool.

The three most common logarithms

• log₁₀(x) — common logarithm. Base 10. log₁₀(1000) = 3 because 10³ = 1000. This is the log behind pH, the Richter scale, and decibel measurements.
• ln(x) — natural logarithm. Base e ≈ 2.71828. It is the natural choice in calculus, continuous growth and decay, and compound-interest formulas because the derivative of ln(x) is simply 1/x.
• log₂(x) — binary logarithm. Base 2. log₂(8) = 3 because 2³ = 8. It appears constantly in computer science and information theory, where it counts bits and balanced-tree depths.

The laws of logarithms

Every base obeys the same algebraic identities, and the quick-reference panel beside the result lists them as you work. They let you break a complicated expression into simpler pieces:

• Product rule: log_b(m × n) = log_b(m) + log_b(n). Multiplication inside becomes addition outside.
• Quotient rule: log_b(m / n) = log_b(m) − log_b(n). Division becomes subtraction.
• Power rule: log_b(m^k) = k × log_b(m). An exponent inside comes out as a multiplier.
• Identities: log_b(1) = 0 for every valid base, and log_b(b) = 1 because b¹ = b.

Change of base

Most physical calculators only have buttons for log₁₀ and ln, so to evaluate a log in any other base you use the change-of-base formula: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). The custom-base mode of this calculator applies exactly that identity internally, so you can compute log₇(50) or log₅(0.2) directly without manual conversion.

Worked example

Suppose you want log₂(80). Using change of base, log₂(80) = ln(80) / ln(2) = 4.38203 / 0.69315 ≈ 6.32193. You can sanity-check that result with the laws: 80 = 16 × 5 = 2⁴ × 5, so log₂(80) = log₂(2⁴) + log₂(5) = 4 + log₂(5). Since log₂(5) ≈ 2.32193, the total is again 6.32193. To reverse the operation, switch to the anti-log panel, choose base 2, and enter 6.32193 — you get back 2^6.32193 ≈ 80, confirming the round trip.

Tips for working with logs

• Estimate before you compute: log₁₀ of any number tells you roughly how many digits it has, so log₁₀(45000) sits between 4 and 5.
• Use the "all logarithms" panel to compare bases at a glance — the same x in base 10, e, and 2 reveals how base choice scales the answer.
• Remember that adding 1 to a base-10 log means multiplying the original number by 10; this is the intuition behind every logarithmic scale.

Common mistakes to avoid

• Splitting a sum incorrectly: log(a + b) is not log(a) + log(b). The product rule applies to multiplication, never to addition.
• Forgetting the domain: there is no real logarithm of 0 or of a negative number, so log(−5) and log(0) are undefined.
• Using base 1: a base of 1 is not allowed because 1 raised to any power is always 1, so it can never reach other values.
• Confusing ln and log: "log" with no base usually means base 10 in engineering but base e in pure mathematics — check the context before trusting a result.