How does the Log Calculator work?

log_b(x) = n means b^n = x Then: Common log: log₁₀(1000) = 3 because 10³ = 1000 Then: Natural log: ln(e) = 1 because e^1 = e Then: Change of base: log_b(x) = log(x) / log(b).

Can you give an example of how to use the Log Calculator?

Example: Input log₁₀(10,000) gives a result of 4 (10⁴ = 10,000).

Is the Log Calculator free to use?

Yes — completely free with no registration, download, or subscription required. All calculations happen instantly in your browser.

How accurate is the Log Calculator?

Our Log Calculator uses verified mathematical formulas and is accurate to multiple decimal places. Results are calculated in real-time using the same methods used by professionals.

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Log Calculator

Logarithm Calculator

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Base

A logarithm is the inverse of exponentiation. log_b(x) answers "to what power must b be raised to get x?" Common logarithm (base 10) is used in science; natural logarithm (base e) in calculus; binary logarithm (base 2) in computer science.

Rule	Formula	Example
Product	log(xy) = log(x) + log(y)	log(100×10) = log(100)+log(10) = 3
Quotient	log(x/y) = log(x) − log(y)	log(100/10) = 2−1 = 1
Power	log(xⁿ) = n·log(x)	log(1000²) = 2×3 = 6
Change of base	log_b(x) = ln(x)/ln(b)	log₂(8) = ln8/ln2 = 3
Inverse	b^(log_b(x)) = x	10^(log₁₀(50)) = 50

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Pro Tip

To estimate log₁₀ of any number mentally: count the digits and subtract 1. log₁₀(8,432) ≈ 3.9 (4 digits → between 3 and 4).

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Did You Know?

The Richter scale, pH scale, decibel scale, and musical octaves are all logarithmic. A pH difference of 1 represents a 10× change in acidity, not a 1-unit linear change.

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