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Log Calculator (Arbitrary Base, Natural & Common Log)

Calculate logarithms for any base, natural log (ln), common log (log10), and binary log with step-by-step formulas.

Quick Answer:A logarithm answers the question: to what power must the base b be raised to equal x? Formally, log_b(x) = y means b^y = x.

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How to Use the Log Calculator (Logarithm Base 10, e & 2)

1Enter the argument number (x) in the first input field.
2Enter the base (b) in the second field (default is 10).
3View the calculated logarithm log_b(x).
4Check the reference values for natural log ln(x), base 10 log, and base 2 log below.

Formula & How It Works

Logarithmic Properties & Change of Base Formula

log_b(x) = ln(x) / ln(b)

By the Change of Base Theorem, log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b), for x > 0 and b > 0, b ≠ 1.

Worked Examples

Common Logarithm Base 10

Input: x = 1000, b = 10

Explanation: 10³ = 1000, therefore log₁₀(1000) = 3.

Result: 3

Binary Logarithm Base 2

Input: x = 64, b = 2

Explanation: 2⁶ = 64, therefore log₂(64) = 6.

Result: 6

Natural Logarithm of e²

Input: x = 7.389056 (e²), b = e

Explanation: ln(e²) = 2.

Result: 2

Frequently Asked Questions

Can a logarithm have a negative result?▼

Yes, when the argument x is between 0 and 1, the logarithm is negative (e.g. log10(0.1) = -1).

Why can't the base be 1?▼

Because 1 raised to any power is always 1, making log₁(x) impossible to define for any number other than 1.

What is the change of base formula?▼

The change of base formula states that log_b(x) = ln(x) / ln(b).

What is Euler's number e?▼

Euler's number e is an irrational mathematical constant approximately equal to 2.718281828.

Is this calculator private?▼

Yes, all logarithmic calculations run directly in your browser with Decimal precision.

Does this support fractional bases?▼

Yes, any positive base other than 1 (such as 0.5 or 1.5) is fully supported.

Quick Reference & Conversion Table

Value (x)	log₁₀(x)	ln(x)	log₂(x)
1	0	0	0
2	0.3010	0.6931	1.0000
10	1.0000	2.3026	3.3219
100	2.0000	4.6052	6.6439
1000	3.0000	6.9078	9.9658

⚠️ Common Pitfalls to Avoid

•Entering zero or negative numbers as the argument, which is undefined in real arithmetic.
•Using base 1 or negative numbers as the base, which violates logarithm definition rules.
•Confusing the natural logarithm (ln, base e) with the common logarithm (log, base 10).

💡 Practical Real-World Uses

•Acoustics and audio engineering (decibel sound pressure levels).
•Seismology and earthquake magnitude measurement on the Richter scale.
•Chemistry and biochemistry (pH acidity scale: pH = -log[H+]).
•Computer science algorithm complexity analysis (e.g. O(log n) binary search).

Natural Log (ln) vs Common Log (log₁₀)

✓Natural log uses Euler's constant e (approx 2.71828) as its base and arises naturally in calculus, biology, and decay models.
✓Common log uses base 10 and aligns with the decimal number system, widely used in engineering metrics.

Last reviewed & updated: 2026-11-01

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