Logarithm (Log & Ln) Calculator
Use our free online Logarithm Calculator to perform fast, accurate calculations with instant formulas, step-by-step arithmetic, and verified reference benchmarks.
Logarithm Calculator
Instant Live Computation • Zero Waiting
Adjust inputs on the left for instantaneous live computation.
How to Calculate: The Logarithm Calculator Formula
Calculates the exponent y to which base b must be raised to produce x. Change of Base: log_b(x) = ln(x) / ln(b).
Step-by-Step Calculation Guide
- Input argument number (x, must be positive).
- Input base (b, default is 10 or Euler's number e ≈ 2.71828).
- Review computed logarithm value.
- Verify by calculating base^result.
Practical Logarithm Calculator Examples
Calculate log10(1,000).
Calculate ln(20).
Core Logarithmic Identity Properties
Official reference values and benchmark classifications based on standard institutional guidelines.
| Log Property | Mathematical Identity | Application Context |
|---|---|---|
| Product Rule | log_b(x × y) = log_b(x) + log_b(y) | Decomposing multipliers |
| Quotient Rule | log_b(x / y) = log_b(x) - log_b(y) | Decibel and audio acoustics |
| Power Rule | log_b(x^k) = k × log_b(x) | Solving compound interest terms |
| Identity Rule | log_b(b) = 1 and log_b(1) = 0 | Universal base baseline |
| Change of Base | log_b(x) = ln(x) / ln(b) | Evaluating arbitrary bases on calculators |
Frequently Asked Questions about Logarithm Calculator
What is the difference between log and ln?
'log' typically denotes common base-10 logarithm log10(x). 'ln' denotes the natural logarithm log_e(x) using Euler's number e (≈ 2.71828).
Why can't you take the log of zero or a negative number?
No real number power exists that turns a positive base into 0 or a negative number. Thus, log(x) is undefined for x <= 0 in real numbers.
Where are logarithms used in real life?
Log scales measure earthquake Richter magnitude, sound decibels (dB), chemical acidity (pH), and data complexity in computer science (O(log n)).
What is binary logarithm (log2)?
Log base 2 calculates information entropy in bits and binary tree search depths in programming.