Logarithm Calculator: Base 2, e, 10 or Custom

A logarithm answers one question: to what power must the base be raised to get the number? If b^y = x, then y = log_b(x), where b is the base and x is the number. For example, log₂(8) = 3 because 2³ = 8.

Most calculators only carry buttons for base 10 (log) and base e (ln), so any other base is found with the change-of-base formula: log_b(x) = ln(x) / ln(b). The logs on the right can use any base at all — this calculator works with the natural logarithm. For a full keypad with dedicated ln, log and power buttons to check a result by hand, try our scientific calculator.

Find log₂(1024). Ask: 2 to what power gives 1024? Since 2¹⁰ = 1024, log₂(1024) = 10. With the change of base: ln(1024) / ln(2) = 6.9315 / 0.6931 = 10.

Two more quick checks: ln(e³) = 3, because e raised to the 3rd power is e³ — the natural log simply hands back the exponent. And log₁₀(1000) = 3, because 10³ = 1000.

Enter the number (x), choose a base, and press Calculate. The number must be greater than zero; the base must be positive and cannot be 1. The calculator shows the result, the exponential check, and the change-of-base working.

Choose the base

• Binary — base 2: the logarithm of computer science, used for bits, doubling and algorithm complexity.
• Natural — base e (≈ 2.71828): written ln(x), the base of calculus and continuous growth.
• Common — base 10: the plain “log” button on most calculators, used across science and engineering.
• Custom: type any positive base except 1 — for example 5 or 7, or the letter e for the natural base.

What you get

• The logarithm: log_b(x), rounded to a sensible number of decimals.
• The exponential check: the base raised to the result, which should equal your number — proof the answer is right.
• The working: ln(x) / ln(b) with both values shown, so you can follow the change of base.

A handful of rules covers almost everything logarithms do. They all assume the same base b, with b > 0 and b ≠ 1, and positive arguments. The product, quotient and power rules come straight from the laws of exponents — a logarithm is the inverse of exponentiation, so multiplication becomes addition and powers become multiplication. The identities are derived in full in the logarithm article on Wikipedia.

Two instant checks fall out of the definition itself: log_b(1) = 0 because b⁰ = 1, and log_b(b) = 1 because b¹ = b.

Three bases appear so often that they have their own names and shorthand:

In the real world, logarithms compress enormous ranges into small, readable ones. The pH scale is −log₁₀ of the hydrogen-ion concentration, so one pH unit means a tenfold change in acidity. Sound intensity in decibels is 10 × log₁₀ of a power ratio, and earthquake magnitude works the same way.

Logarithms also run exponential growth backwards. If money grows at a fixed rate, the logarithm answers how long until it doubles — the same growth modeled by our compound interest calculator. In computing, binary logarithms measure how many times you can halve a problem, which is why efficient search algorithms are described as O(log n).

The domain rules are not arbitrary — they follow from what exponentiation can actually produce.

The number (x) must be positive

A positive base raised to any real power stays positive, so no exponent can ever produce zero or a negative number. That is why log_b(0) and log_b(−5) are undefined. As x slides toward zero from above, the logarithm dives toward negative infinity — it never lands.

The base must be positive and cannot be 1

A negative base would make b^y flip between positive and negative values in a way that breaks the definition for real numbers. And base 1 is useless: 1 raised to any power is 1, so “1 to what power equals x?” has no answer for any x except 1 (where every answer works). This calculator rejects both cases with a clear message instead of showing NaN.