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Log Calculator

Calculate logarithms in any base with step-by-step solutions. Includes natural log (ln), common log (log₁₀), and custom base logarithms.

Logarithm Results

logₐ(x) 0
Antilog (aˣ) 0

Comparison Table

Base Notation Value

Step-by-Step Calculation

How to Use the Log Calculator

  1. Enter the number (x) you want to find the logarithm of.
  2. Select the base: common log (10), natural log (e), binary log (2), or custom.
  3. If custom base is selected, enter the base value.
  4. Click "Calculate Logarithm" to see the result.
  5. Review the comparison table and step-by-step calculation.

Formulas Used

Logarithm Definition: logₐ(x) = y means aʸ = x

Change of Base Formula:

logₐ(x) = ln(x) / ln(a) = log₁₀(x) / log₁₀(a)

Common Logarithms:

  • log₁₀(x) = common logarithm (base 10)
  • ln(x) = natural logarithm (base e ≈ 2.71828)
  • log₂(x) = binary logarithm (base 2)

Antilogarithm: antilogₐ(y) = aʸ

Frequently Asked Questions

What is a logarithm?

A logarithm answers the question: "To what power must we raise the base to get this number?" For example, log₁₀(100) = 2 because 10² = 100. Logarithms are the inverse operation of exponentiation.

What's the difference between ln and log?

ln (natural log) uses base e (Euler's number ≈ 2.71828), while log typically means base 10 (common log). Natural logs are fundamental in calculus and continuous growth, while common logs are used in engineering and pH calculations.

Can I take the log of a negative number?

No, logarithms of negative numbers are undefined in real numbers. The result would be a complex number with an imaginary part. For real logarithms, the input must be positive. Our calculator requires positive inputs.

What is log of 1?

The logarithm of 1 is always 0 in any base, because any number raised to the power of 0 equals 1. logₐ(1) = 0 for any valid base a > 0, a ≠ 1.

When would I use logarithms in real life?

Logarithms are used in: measuring sound intensity (decibels), earthquake magnitude (Richter scale), pH levels, radioactive decay, population growth models, compound interest calculations, and many scientific and engineering applications.