Log Calculator: log, ln, log2, Any Base and Antilog

How to use the calculator

Pick a mode. Logarithm answers “b to what power gives x?” Antilog computes b^y.
Choose the base: 10, e, 2, or a custom base. A custom base must be greater than 0 and not equal to 1.
Enter x (log mode, must be greater than 0) or y (antilog mode, any number from −1,000,000 to 1,000,000).

In log mode the calculator also shows ln x, log₁₀ x and log₂ x, and for every base except e it writes out the change-of-base step with the numbers filled in. The check line shows the inverse in both modes.

How the calculator works

A logarithm is an exponent. For x > 0, b > 0 and b ≠ 1:

log_b(x) = y means b^y = x

Bases 10, e and 2 use the built-in log functions. Any other base uses the change-of-base formula:

log_b(x) = ln(x) ÷ ln(b)

Results are rounded to 12 significant digits. That removes floating-point noise: log₃(81) shows 4, not 4.000000000000001. MDN notes the same effect in JavaScript, where dividing logs to get log₁₀(1,000) returns 2.9999999999999996.

In antilog mode the calculator returns b^y (e^y for base e). For results beyond about 10^±300, near the limits of ordinary floating-point numbers, it builds the scientific form from y × log₁₀(b): 10^500.5 = 3.16227766 × 10⁵⁰⁰.

Worked examples

Logarithm	Result	Check
log₁₀(1,000)	3	10^3 = 1,000
log₁₀(2)	0.301029995664	10^0.301029995664 = 2
ln(10)	2.30258509299	e^2.30258509299 = 10
log₂(1,024)	10	2^10 = 1,024
log₀.₅(8)	−3	0.5^(−3) = 8

1. A custom base, step by step

For log₃(81), the change-of-base line reads ln(81) ÷ ln(3) = 4.39444915467 ÷ 1.09861228867 = 4, because 3⁴ = 81.

2. Doubling time

How many years does money take to double at 5% a year? Solve 1.05^t = 2, which is log₁.₀₅(2) = 14.2066990829. At 7%, log₁.₀₇(2) = 10.2447683511.

3. Counting digits

log₁₀(123,456,789) = 8.09151497717. Drop the decimals and add 1 to get the number of digits, 9. In general a positive whole number n has ⌊log₁₀ n⌋ + 1 digits.

Where logarithms show up

Logarithmic scales compress huge ranges into small numbers, so each step of 1 means a factor of 10. The USGS describes pH in logarithmic units: water at pH 5 is ten times more acidic than water at pH 6. For earthquake magnitude, the USGS says each whole number is a tenfold increase in measured amplitude and about 32 times more energy release. Because log₁₀(10,000) = 4, a difference of 4 magnitude units means 10,000 times the amplitude.

Frequently asked questions

Does “log” mean base 10 or the natural log?

It depends on who wrote it. OpenStax's College Algebra uses log x with no base for the common logarithm, base 10, while ln x always means base e. Software often differs: MDN documents that JavaScript's Math.log returns the natural logarithm. Check which base a formula or tool means before you mix results.

What is ln?

ln is the natural logarithm, the logarithm to base e, where e is about 2.71828182846 (the antilog of 1 in base e). ln(10) = 2.30258509299. Natural logs appear in continuous growth and decay, calculus and statistics.

Why can't I take the log of 0 or a negative number?

A positive base raised to any real power is always positive, so no real exponent gives 0 or a negative number. The real logarithm is defined only for x > 0. Base 1 is excluded too, because 1 to any power is 1.

What is an antilog?

The antilog undoes a logarithm: antilog₁₀(3) = 10^3 = 1,000. It also works for negative and fractional exponents: 10^(−7.4) = 3.981071706 × 10⁻⁸.

How do I calculate a log with a different base?

Divide two logs in the same base: log_b(x) = ln(x) ÷ ln(b), or log₁₀(x) ÷ log₁₀(b). Either gives the same answer. Choose Custom base and the calculator shows this step with your numbers.

Can the result be negative?

Yes. With a base greater than 1, any x between 0 and 1 has a negative log: log₁₀(0.001) = −3. With a base between 0 and 1 the signs flip, so log₀.₅(8) = −3.

Sources

This calculator is a math tool for learning and checking work. Results are rounded to 12 significant digits.