Log Calculator
Calculate logarithms with any base. Find log base 10, natural log (ln), or custom bases instantly.
Logarithm Calculation
(Use e for natural log)
Logarithm Results
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log₁₀(100)
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base e
0
base 10
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base 2
💡 Definition: The logarithm of a number is the exponent to which the base must be raised to produce that number. If b^y = x, then log_b(x) = y.
What is a Logarithm?
A logarithm is the inverse operation of exponentiation. It answers the question: "To what power must I raise the base to get this number?"
If b^y = x, then log_b(x) = y
Example: 10^2 = 100, so log₁₀(100) = 2
Example: 10^2 = 100, so log₁₀(100) = 2
Types of Logarithms
| Type | Base | Notation | Common Use |
|---|---|---|---|
| Common Log | 10 | log(x) or log₁₀(x) | Science, engineering, pH levels |
| Natural Log | e (2.718...) | ln(x) or log(x) | Calculus, physics, finance |
| Binary Log | 2 | log₂(x) | Computer science, information theory |
Logarithm Rules
- Product Rule: log_b(xy) = log_b(x) + log_b(y)
- Quotient Rule: log_b(x/y) = log_b(x) - log_b(y)
- Power Rule: log_b(x^n) = n × log_b(x)
- Change of Base: log_b(x) = log_k(x) / log_k(b)
- Identity: log_b(b) = 1
- Zero Rule: log_b(1) = 0
Real-World Applications
- pH Scale: Measures acidity (pH = -log[H+]).
- Richter Scale: Measures earthquake magnitude logarithmically.
- Decibels: Measures sound intensity (dB = 10 × log(I/I₀)).
- Compound Interest: Used to solve for time in financial formulas.
- Algorithms: Binary search has O(log n) time complexity.
References
- Khan Academy. "Introduction to Logarithms." 2026.
- Purplemath. "Logarithm Rules." 2026.