Mathematics

Logarithm Calculator

Value (x)

Base

What is Logarithm Calculator?

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A logarithm calculator computes log base 10, natural log (ln), log base 2, and any other base. Logarithms are the inverse of exponentiation.

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Formula

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f(x)log_b(x) = y means b^y = x; Change of base: log_b(x) = log_10(x) / log_10(b)

Variable Legend

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Symbol	Name	Unit	Description
x	Number	—	—
b	Base	—	Usually 10, e, or 2
y	Logarithm result	—	—

How to Logarithm Calculator

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1logₐ(x) = y means aʸ = x
2log₁₀(x): common logarithm
3ln(x): natural log, base e ≈ 2.71828
4Change of base: logₐ(x) = ln(x)/ln(a)

Worked Examples

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Example 1

Given:log₁₀(1000)

Result:3 (because 10³ = 1000)

Example 2

Given:ln(e²)

Result:2

Real-World Applications

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Scientific and engineering calculations

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Earthquake magnitude (Richter scale)

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Sound intensity (decibels)

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pH calculations in chemistry

Logarithm Properties

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Property	Formula
Product rule	log(ab) = log(a) + log(b)
Quotient rule	log(a/b) = log(a) − log(b)
Power rule	log(aⁿ) = n×log(a)
Change of base	logₐ(x) = log(x)/log(a)
log(1)	Always = 0, any base

Frequently Asked Questions

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What's the difference between log, ln, and log₁₀?

log (no base) = log₁₀. ln = natural log (base e ≈ 2.718). log₂ = binary logarithm.

When do I use logarithms?

Solving exponential equations, measuring scales (pH, Richter, decibels), growth/decay rates.

Why is log₁₀ of 1000 equal to 3?

10³ = 1000, so log₁₀(1000) = 3. Logarithm is the inverse of exponentiation.

References

›Logarithms — Khan Academy

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Logarithms vs Exponents

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Reviewed October 2026

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Logarithm Calculator