What is the difference between log and ln?

log (without a base specified) typically means log base 10. ln is the natural logarithm, which uses base e ≈ 2.71828. Both are available in this calculator.

Why can't I take the log of a negative number or zero?

Logarithms are only defined for positive numbers. log(0) approaches negative infinity and log of a negative number has no real solution.

What is a natural logarithm used for?

The natural log (ln) appears naturally in continuous growth and decay, probability, calculus, and physics. It's the inverse of the exponential function e^x.

How do I change the base of a logarithm?

Use the change-of-base formula: log_b(x) = log(x) / log(b) = ln(x) / ln(b). This calculator applies this formula automatically for any custom base.

What is log base 2 used for?

Log base 2 is fundamental in computer science and information theory. It tells you how many bits are needed to represent a number — for example, log₂(256) = 8 means 8 bits.

Math

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Log Calculator - Logarithm Base 10 Natural Ln Online

Calculate logarithms with any base — log base 10, natural log (ln), log base 2, or any custom base.

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How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

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In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

A logarithm calculator finds the logarithm of a number for any base — including base 10 (common log), base e (natural log, ln), base 2 (binary log), or any custom base you specify. Logarithms are the inverse of exponentiation and appear throughout science, engineering, music, and information theory.

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What is Log Calculator - Logarithm Base 10 Natural Ln Online?

How to use it

1️⃣ Select the base from the dropdown: base 10, base e (ln), base 2, or custom.
2️⃣ If you choose custom, enter your desired base in the field that appears.
3️⃣ Enter the number you want to take the log of in the 'of' field.
4️⃣ The result shows instantly, along with log₁₀, ln, and log₂ values for the same number.

Formula

log_b(x) = ln(x) / ln(b). Special cases: log₁₀(x) = log(x); log_e(x) = ln(x); log₂(x) = log(x) / log(2). The input x must be greater than 0.

💡 See it in action — a real example

log₁₀(100) = 2 because 10² = 100. ln(e) = 1 because e¹ = e. log₂(8) = 3 because 2³ = 8. For log₃(81), use the custom base field to get log₃(81) = 4.

❓ Common questions

What is the difference between log and ln?: log (without a base specified) typically means log base 10. ln is the natural logarithm, which uses base e ≈ 2.71828. Both are available in this calculator.
Why can't I take the log of a negative number or zero?: Logarithms are only defined for positive numbers. log(0) approaches negative infinity and log of a negative number has no real solution.
What is a natural logarithm used for?: The natural log (ln) appears naturally in continuous growth and decay, probability, calculus, and physics. It's the inverse of the exponential function e^x.
How do I change the base of a logarithm?: Use the change-of-base formula: log_b(x) = log(x) / log(b) = ln(x) / ln(b). This calculator applies this formula automatically for any custom base.
What is log base 2 used for?: Log base 2 is fundamental in computer science and information theory. It tells you how many bits are needed to represent a number — for example, log₂(256) = 8 means 8 bits.

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