About the Logarithm Calculator
This logarithm calculator answers the question "what power do I raise the base to in order to get this number?" Enter a positive number, pick the base — 10 (common log), e (natural log, ln), 2 (binary log) or any custom base — and it returns the logarithm together with ln, log₁₀ and log₂ of the same number so you can compare them at a glance.
Switch the mode to antilog to go the other way: it raises the base to your exponent, which is how you undo a logarithm (for example 10^2.5 or e^1.2). That is handy for decibels, pH, the Richter scale, compound growth and any formula where an answer was reported as a log.
Logarithms are only defined for positive numbers, and the base must be positive and not equal to 1. If you enter something outside those rules the calculator tells you why instead of returning a meaningless number.
With the default inputs, the result is 3. Change any value above to recalculate instantly.
How to use the logarithm calculator
- 1Choose Logarithm to find log_b(x), or Antilog to compute b raised to a power.
- 2Enter the number (or, for antilog, the exponent).
- 3Pick base 10, e, 2, or choose Custom base and type your own.
- 4Read the result and the ln, log₁₀ and log₂ values shown alongside it.
- 5Use the check line to confirm that raising the base to the answer gives back your number.
Formula and method
A logarithm is the inverse of an exponent: log_b(x) is the power y that the base b must be raised to in order to produce x. So log₁₀(1000) = 3 because 10³ = 1000, and log₂(8) = 3 because 2³ = 8.
Calculators and computers only need one logarithm to compute all the others. This tool uses the change-of-base rule, dividing the natural log of x by the natural log of the base. The antilog simply reverses the operation by computing b^x. Logs are defined only for x > 0 with a base b > 0 and b ≠ 1.
- x
- The number (argument), must be positive
- b
- The base: 10, e ≈ 2.71828, 2 or any positive number except 1
- y
- The logarithm — the exponent that turns b into x
- ln
- Natural logarithm, base e
Worked examples
Common log of 1,000
Because 10³ = 1000, log₁₀(1000) is exactly 3. The natural log of 1000 is about 6.9078 and its base-2 log is about 9.9658, which tells you 1000 sits just under 2¹⁰ = 1024.
Natural log of 50
ln(50) ≈ 3.9120, meaning e raised to about 3.912 equals 50. The common log of 50 is about 1.699, i.e. 50 is between 10¹ and 10².
Log base 5 of 200 (change of base)
Using change of base, log₅(200) = ln(200) ÷ ln(5) = 5.2983 ÷ 1.6094 ≈ 3.2920. Check: 5^3.292 ≈ 200.
Antilog: 10 to the power 2.5
The antilog of 2.5 in base 10 is 10^2.5 = √100000 ≈ 316.23. This is how you convert a value reported on a log scale back into ordinary units.
Frequently asked questions
What is the difference between log and ln?+
"log" usually means the common logarithm with base 10, while "ln" is the natural logarithm with base e ≈ 2.71828. In many programming languages and higher-level math texts, however, log() means the natural log, so check the convention.
How do I calculate a log with a different base?+
Use the change-of-base formula: log_b(x) = ln(x) ÷ ln(b), or equivalently log₁₀(x) ÷ log₁₀(b). For example log₃(81) = ln 81 ÷ ln 3 = 4.3944 ÷ 1.0986 = 4.
Why can’t I take the log of zero or a negative number?+
No real power of a positive base can ever equal zero or a negative number, so log(0) and log of negatives are undefined in real numbers. As x approaches 0 from above, log(x) heads toward negative infinity.
What is an antilog?+
The antilog reverses a logarithm by raising the base to the given power. The base-10 antilog of 2 is 10² = 100, and the natural antilog of 1 is e¹ ≈ 2.71828.
What are the main logarithm rules?+
Product rule: log(ab) = log a + log b. Quotient rule: log(a/b) = log a − log b. Power rule: log(aⁿ) = n·log a. Also log_b(1) = 0 and log_b(b) = 1 for any valid base.