Logarithm & Natural Log (ln) Calculator
Evaluate common log ($log_{10}$), natural log ($ln$), binary log ($log_2$), and custom base logarithms instantly.
2.000
Base 104.605
Base e ≈ 2.7186.644
Base 22.861
Base 5Change of Base Verification Formula:
What is a Logarithm?
A logarithm is the mathematical inverse function to exponentiation. It answers the fundamental question: "To what power must the base be raised in order to produce number $x$?" For example, because $10^2 = 100$, the common logarithm is $log_{10}(100) = 2$. Logarithmic scales are essential across science for measuring earthquake intensity (Richter scale), acoustic loudness (Decibels), and chemical acidity (pH scale).
Key Capabilities
- Simultaneous Multi-Base Display: Evaluates Common ($log_{10}$), Natural ($ln$), Binary ($log_2$), and arbitrary custom bases simultaneously.
- Change of Base Precision: Applies the exact algebraic ratio ($log_b(x) = rac{ln x}{ln b}$) for arbitrary fractional bases.
- Domain Protection: Flags undefined domains immediately when $x le 0$ or base $b le 0$ or $b = 1$.
- Scientific Precision: Delivers high floating-point resolution with clean formatting.
How to Use the Logarithm Calculator
- Enter the target value $x$ (must be strictly greater than 0).
- Enter your custom base $b$ (e.g. 2, 5, 10, or any positive real number other than 1).
- Read off the corresponding logarithmic power values across all four standard bases.
Frequently Asked Questions
Why can't the base of a logarithm equal 1?
Because 1 raised to any power is always 1 ($1^y = 1$). You can never produce any number other than 1, and for $x=1$ the exponent is ambiguous and indeterminate.
What is the value of log(1) for any base?
For any valid base $b$, $log_b(1) = 0$, because any non-zero number raised to the power of 0 equals 1 ($b^0 = 1$).
Where is binary log (log2) used in computer science?
Binary logarithms ($log_2$) measure information entropy (bits), binary tree search complexities ($O(log n)$ in binary search and quicksort), and memory addressing.
Is this tool free?
Yes, it is 100% free with unlimited calculations.