Logarithm Calculator (any base)

A logarithm asks: what power turns the base into the argument? log₂(1024) is 10 because 2¹⁰ is 1024. The base, the argument and that power form a small system, and this calculator solves for whichever one you leave out — the log itself, the argument behind a known log, or the base that would make a pair work.

The base field takes any positive number except 1, including the literal e for a natural log, because every answer is computed by change of base through ln rather than a lookup limited to 2, 10 and e. Alongside the result, the panel restates the argument’s logarithm in all three standard bases so you can read across.

Running the Calculator in Each Direction

1. Choose Solve for: Result, Argument, or Base. The two fields you are not solving for become the inputs.
2. Enter the knowns. Load Sample opens on base 2 and argument 1024, and the answer is a clean 10. The three base preset chips fill 10, e or 2 in one click.
3. To go backwards, switch to Argument x and enter a base and a result: base 2 with result 10 returns 1024. Base a takes an argument and a result and returns the base that fits.
4. Read the standard bases panel for log₁₀, ln and log₂ of the same argument, and the change-of-base line that shows the arithmetic. Press Copy for the headline value.

The Change-of-Base Method

• Every result uses log₀(x) = ln(x) / ln(a). That is why an arbitrary base like 7 or 1.5 works exactly as well as 10.
• Solving for the argument is x = ar; solving for the base is a = x(1/r).
• The result card also writes the answer back as an exponent equation, so a log of 10 from base 2 and argument 1024 is shown as 210 = 1024 underneath.
• A non-integer case stays exact to the display precision: log₁₀(500) reads 2.698970004, and the base-2 row of the same argument reads 8.965784285.

What This Rounds and Where Logs Are Undefined

• Values are shown to 10 significant figures, with trailing zeros trimmed, so log₂(1024) is a plain 10 rather than 10.00000000.
• The argument must be positive. Zero or a negative stops with “The argument must be greater than 0 — the logarithm of zero or a negative number is undefined.”
• The base must be positive and not 1, since 1 to any power is always 1 and could never reach a different argument.
• When solving for the base, a result of 0 is refused (every base to the power 0 is 1, so the base is unconstrained), and an argument of exactly 1 with a non-zero result has no answer, because log₀(1) is 0 in every base.
• Arithmetic is IEEE-754 double precision, so a value far outside everyday range (a base near 1, an argument past 10³⁰⁰) can lose accuracy in the last figures or overflow to infinity.