Logarithm Calculator
Calculate logarithms for any base (10, e, 2, or custom) and anti-logarithms. Includes natural log (ln), common log, and binary log calculations.
Log & Anti-Log Calculator
Calculate logs for any base or inverse exponentiation
The number you want to calculate the log or anti-log for
Select the base for the logarithm
Choose whether to find the logarithm or the inverse (anti-log)
Your Log Calculation
Result, formulas, and alternative bases
Enter a number, select your base and calculation type, then click Calculate to see the logarithm or anti-logarithm result.
Common Logarithm Benchmarks
Quick reference guide for the logarithms of common numbers across different bases.
| Number (x) | Common Log (log10) | Natural Log (ln) | Binary Log (log2) |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.3010 | 0.6931 | 1 |
| 10 | 1 | 2.3026 | 3.3219 |
| 100 | 2 | 4.6052 | 6.6439 |
| 1000 | 3 | 6.9078 | 9.9658 |
| e (2.718) | 0.4343 | 1 | 1.4427 |
Logarithm FAQ
Everything you need to know about logarithms, natural logs, and exponential calculations.
A logarithm is the exponent to which a base must be raised to produce a given number. For example, the base 10 logarithm of 100 is 2, because 10 raised to the power of 2 equals 100. It is written as log10(100) = 2.
The term ‘log’ without a specified base usually refers to the common logarithm, which has a base of 10 (log10). The term ‘ln’ stands for the natural logarithm, which has a base of ‘e’ (Euler’s number, approximately 2.71828). The natural logarithm is widely used in calculus, physics, and financial modeling.
To calculate a logarithm with a custom base ‘b’, you can use the Change of Base Formula: log_b(x) = ln(x) / ln(b) or log_b(x) = log10(x) / log10(b). This formula allows you to find the log of any number using any valid base by dividing the natural log (or common log) of the number by the natural log (or common log) of the base.
An anti-logarithm (or inverse logarithm) is the inverse operation of a logarithm. It is essentially exponentiation. If log_b(x) = y, then the anti-log of y with base b is x, which is written as b^y = x. For example, the anti-log of 2 with base 10 is 100, because 10^2 = 100.
No, in the realm of real numbers, you cannot take the logarithm of zero or a negative number. The logarithm function is only defined for strictly positive real numbers. If you attempt to calculate the log of a negative number, the result is a complex number, which requires imaginary numbers (involving ‘i’, the square root of -1) to express.
The three fundamental rules of logarithms are: 1) Product Rule: log_b(x * y) = log_b(x) + log_b(y). 2) Quotient Rule: log_b(x / y) = log_b(x) – log_b(y). 3) Power Rule: log_b(x^n) = n * log_b(x). These rules are essential for simplifying complex logarithmic expressions and solving exponential equations.