How do I calculate a logarithm?

Divide the natural log of the value by the natural log of the base. For log base 2 of 8, that is ln(8) divided by ln(2), which is 2.0794 divided by 0.6931, giving exactly 3.

What is the change of base formula?

log base b of x equals log base a of x divided by log base a of b, for any base a you like. Using natural logs, it becomes ln(x) divided by ln(b). This is why any logarithm can be converted to any other base without a special formula.

What is the difference between common log and natural log?

Only the base. The common logarithm uses base 10 and is written log with no subscript. The natural logarithm uses base e, approximately 2.71828, and is written ln. They are the same calculation with a different base, and this calculator returns both from one input.

Why can a logarithm not be zero or negative?

It can. A value of 1 gives a log of 0, and any value between 0 and 1 gives a negative log, because raising a base to a negative power produces a fraction. What cannot exist is a real log of a zero or negative value.

The inverse of a logarithm. Given log base b of x equal to n, the antilog finds x by raising b to the power of n. The Antilog button on this calculator does exactly that and shows the log of the result as a check.

Why is log base 1 undefined?

Because 1 raised to any power is 1, so there is no exponent that turns 1 into anything except 1. The logarithm is searching for that exponent, and no such exponent exists for any other value.

How do I solve 2 to the power of x equals 100?

Take the log of both sides. x multiplied by log base 2 of 2 equals log base 2 of 100, and log base 2 of 2 is 1, so x equals log base 2 of 100, which is about 6.644. Use the Solve for value button on this calculator for the direct route.

What are logs used for in real life?

Anywhere a ratio is measured on a scale that matches human perception. Decibels for sound, pH for acidity, the Richter scale for earthquakes and stellar magnitude for brightness all use logarithms, which is why each doubling on those scales has a consistent physical meaning.

Why does my calculator give 2.9999999999999996 instead of 3?

Floating point representation of numbers cannot store 8 and 2 as exact fractions of their true values, so dividing their natural logs leaves a tiny remainder. The mathematical answer is 3, and this calculator snaps results that fall within a very small tolerance of a whole number so the exact case displays correctly.

How precise is a logarithm result?

The precision of the answer matches the precision of the input. Doubling the number of decimal places in your input adds about one decimal place of reliability to the output, and no more. Rounding the result further than the input supports implies accuracy you do not have.

How to Use the Logarithm Calculator

This calculator returns your logarithm in the base you choose and, at the same time, the common log, natural log and binary log of the same number, plus an antilog and a solve for value mode.

Step-by-Step Instructions

1 Enter the value you want the logarithm of. It must be greater than zero, since a real logarithm of zero or a negative number does not exist.
2 Enter the base. Use 10 for common log, e or 2.71828 for natural log, or 2 for binary log. Any positive base other than 1 works.
3 Read your answer alongside the common, natural and binary versions, which are returned in the same pass so you do not have to re-enter anything.
4 Press Antilog to reverse the operation and recover the original value from a logarithm result.
5 Press Solve for value to answer the reverse question, which power of a base gives this number.

Understanding the Inputs

Value

The number you are taking the logarithm of, or the exponent when using the Antilog and Solve modes. Must be greater than zero for a log calculation.

How to find it: Any positive number. A power of the base gives a whole number answer, such as 1000 for base 10 or 1024 for base 2. A number between zero and one gives a negative answer.

Why it matters: Logarithms invert exponentiation, so the value is the output of a power and the log recovers its exponent. Zero and negatives have no real log, which the calculator explains rather than failing silently.

Type: number · Default: 1000

Base

The number being raised to a power. Any positive real except 1. Use 10, e, or 2 for the three common conventions.

How to find it: Most school work uses 10 or e. Computer science uses 2 exclusively. Anything between 0 and 1 also works and produces negative results, for example log base 0.5 of 8 is minus 3.

Why it matters: The base is the denominator of the change of base formula, which is the entire reason a calculator can do any base at all. Base 1 is excluded because every power of it is 1, so no exponent could ever be found.

Type: number · Default: 10

Tips & Best Practices

✓ Use the change of base formula rather than memorising separate buttons for each base.
✓ Check any answer against the fact that log of 1 is 0 in every base.
✓ Treat results within a tiny tolerance of a whole number as that whole number when your inputs are exact.
✓ For problems like 2 to the power of x equals 100, use the Solve for value button instead of algebra.

Try the Logarithm Calculator

by CalculatorPro Tools · Updated 10-12-2026