Logarithm Calculator
Calculate common, natural, binary, or custom-base logarithms, or evaluate an antilogarithm by raising a base to an exponent.
Logarithm and antilogarithm formulas
A logarithm answers the question “what exponent produces this number?” The statement logb(x)=y is equivalent to by=x, where x>0, b>0, and b≠1.
The common logarithm uses base 10 and is written log(x). The natural logarithm uses base e and is written ln(x). The binary logarithm uses base 2 and is written log2(x).
Any valid base can be calculated with the change-of-base formula: logb(x)=ln(x)/ln(b). The inverse operation, often called an antilogarithm, is x=by.
Checked examples
| Operation | Equivalent exponential form | Result |
|---|---|---|
| log₁₀(1000) | 10³=1000 | 3 |
| log₂(32) | 2⁵=32 | 5 |
| ln(e) | e¹=e | 1 |
| log₃(81) | 3⁴=81 | 4 |
| log₀.₅(8) | 0.5⁻³=8 | −3 |
| log₁₀(0.01) | 10⁻²=0.01 | −2 |
| log₇(1) | 7⁰=1 | 0 |
| Antilog base 10 of 3 | 10³ | 1000 |
| Antilog base 2 of −3 | 2⁻³ | 0.125 |
| Antilog base e of 1 | e¹ | e≈2.71828182846 |
How to use the calculator
- Select logarithm or antilogarithm.
- For a logarithm, enter the positive number x. For an antilogarithm, enter the exponent y.
- Enter 10, e, 2, or another positive base other than 1.
- Select the calculate button and review the result, alternative logarithms, and inverse check.
Limits and interpretation
- A real logarithm requires x>0.
- The base must be positive and cannot equal 1.
- Numbers and bases may be entered in scientific notation and may not exceed 1e100.
- Antilogarithm exponents are limited to a magnitude of 10,000; results that overflow or underflow are rejected.
- Displayed results use approximately 14 significant digits and JavaScript double-precision arithmetic.
- This calculator evaluates real logarithms. It does not return complex logarithm branches for negative inputs.
Frequently asked questions
What is the difference between log and ln?
In most calculator contexts, log means base 10 and ln means base e. Both are logarithms and differ only by their base.
Why can the base not equal 1?
Every power of 1 equals 1, so it cannot produce other positive numbers and does not define an invertible logarithm.
Can a logarithm be negative?
Yes. With a base greater than 1, numbers between 0 and 1 have negative logarithms. For example, log₁₀(0.01)=−2.
What is an antilogarithm?
It reverses a logarithm. If logb(x)=y, then the antilogarithm of y in base b is by=x.
Related calculators
Use the Scientific Notation Calculator for powers of ten, the Square Root Calculator for fractional powers and radicals, or the Quadratic Formula Calculator for polynomial roots.