Log Calculator (Logarithm)

Logarithms answer the question: to what power must a base be raised to produce a given number? They turn multiplication into addition and powers into multiplication, which made them indispensable for calculation before electronic calculators and keeps them central to science today.

What is Log?

The logarithm is the inverse of exponentiation. If bʸ = x, then log_b(x) = y. Since 10² = 100, log₁₀(100) = 2; since 2⁵ = 32, log₂(32) = 5. The base b must be positive and not equal to 1, and the number x must be positive, because no real power of a positive base gives zero or a negative number.

Two bases are used so often they have their own notation. The common logarithm, base 10, is written log or log₁₀ and counts powers of ten; it underlies the decibel scale, pH and the Richter magnitude. The natural logarithm, base e ≈ 2.71828, is written ln and arises naturally in calculus, continuous growth and decay, and probability. The binary logarithm, base 2, is common in computing. With the default inputs, ln(100) ≈ 4.60517.

Basic Log Rules

Logarithm rules mirror the exponent laws. They hold for any valid base b and positive numbers x and y.

Product rule: log_b(xy) = log_b(x) + log_b(y). The log of a product is the sum of the logs. log₁₀(1,000 × 100) = 3 + 2 = 5.

Quotient rule: log_b(x / y) = log_b(x) − log_b(y). log₂(64 ÷ 4) = 6 − 2 = 4.

Power rule: log_b(xⁿ) = n log_b(x). log₁₀(10⁴) = 4 × 1 = 4. This rule is how logarithms solve equations with an unknown exponent: from 2ⁿ = 50 we get n log 2 = log 50, so n ≈ 5.644.

Change of base: log_b(x) = log_k(x) ÷ log_k(b) for any convenient base k. Calculators usually provide only ln and log₁₀, so log₇(343) is computed as ln 343 ÷ ln 7 = 3. This calculator uses the same identity internally.

Special values: log_b(1) = 0 for every base, because b⁰ = 1, and log_b(b) = 1, because b¹ = b. Logs of numbers between 0 and 1 are negative: log₁₀(0.01) = −2. Finally, b^(log_b x) = x and log_b(bˣ) = x, expressing that the logarithm and the power undo each other.

Frequently asked questions