Log Calculator
Calculate logarithms in base 10 (common log), base 2 (binary log), natural logarithm (ln), or any custom base instantly.
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Formula
log_b(x) = ln(x) / ln(b)
Any logarithm can be computed using the change-of-base formula: divide the natural log (or any log) of the value by the natural log of the base. This is how calculators compute logs in arbitrary bases — using only one log function internally.
How to use the Log Calculator
- 1
Enter your value (x)
- 2
Enter your base
Use 2.718 for natural log
- 3
Read your results instantly
Results update in real time as you type.
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What logarithms actually mean
A logarithm answers the question: what exponent do I raise the base to, in order to get this number? log₁₀(1000) = 3 because 10³ = 1000. ln(e) = 1 because e¹ = e. log₂(64) = 6 because 2⁶ = 64.
Logarithms are the inverse of exponential functions, just as subtraction inverts addition. They 'undo' exponentiation, transforming multiplicative relationships into additive ones — which is why slide rules (mechanical logarithm machines) enabled complex calculations for centuries before computers.
The three common bases
Base 10 (common logarithm, log₁₀): used in chemistry (pH scale), seismology (Richter scale), acoustics (decibels), and scientific notation. Every order-of-magnitude change equals 1 unit on a log₁₀ scale.
Base e (natural logarithm, ln): e ≈ 2.71828... is the base of continuous exponential growth. It appears in calculus, statistics, physics, and finance. The natural log is the 'canonical' log in higher mathematics.
Base 2 (binary logarithm, log₂): fundamental in computer science. log₂(n) tells you how many bits are needed to represent n distinct values.
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Logarithmic scales in everyday life
Many phenomena humans perceive follow logarithmic scales. Hearing: the decibel scale is logarithmic — 20 dB is 10× more intense than 10 dB, not 2× more. Vision: one stop in photography doubles or halves the light, matching how the eye adapts.
Earthquakes: each magnitude unit on the Richter scale represents 10× more ground motion amplitude. A magnitude 7 earthquake is 10× stronger than magnitude 6, and 100× stronger than magnitude 5. This is why the difference between a magnitude 5 and magnitude 7 is catastrophic, not merely twice as bad.
Tips & Insights
Log rules for quick simplification
Key identities: log(a×b) = log(a) + log(b). log(a/b) = log(a) − log(b). log(aⁿ) = n×log(a). These let you break complex log expressions into simpler parts.
ln and e are inverses
e^(ln x) = x and ln(e^x) = x. These inverse operations cancel out. This is useful when solving exponential equations: take ln of both sides to bring the exponent down.
Worked Examples
pH calculation
log₁₀(0.001) = −3. pH = −log₁₀[H⁺] = −(−3) = 3. This is the pH of vinegar — moderately acidic.
Doubling time calculation
log₁.₀₇(2) ≈ 10.24 years to double at 7% annual growth. This confirms the Rule of 72: 72 ÷ 7 = 10.3 years.
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Frequently Asked Questions
What is a logarithm?
A logarithm is the inverse of exponentiation. log_b(x) = y means b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000.
What is the natural logarithm?
The natural logarithm (ln) uses base e ≈ 2.71828... It is fundamental in calculus and naturally describes continuous growth and decay processes.
What is log base 2 used for?
Log base 2 is essential in computer science. It tells you how many binary digits (bits) are needed to represent a value, and appears in algorithm complexity analysis (binary search is O(log₂n)).
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