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Log Calculator

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Find the logarithm of any number, with any base - including natural log (base e) and common log (base 10).

Log Calculator

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log₁₀(32)
1.50515
Logarithms are only defined for positive numbers.

The formula

log_b(x) = the power you raise b to, to get x Calculated via: log_b(x) = ln(x) / ln(b) (change of base formula)
Example

log base 2 of 32: since 2⁵ = 32, log₂(32) = 5. The common log (base 10) of 32 is approximately 1.50515.

Step-by-step guide

  1. Enter the number you want to find the logarithm of.
  2. Choose a base - base 10 (common log), base e (natural log), base 2, or any custom base.
  3. Read the result.

What a logarithm actually asks

A logarithm answers a simple question phrased backward from what you might expect: "what power do I need to raise this base to, to get this number?" Log base 2 of 32 asks "2 raised to what power equals 32?" and the answer is 5, since 2⁵=32. This is exactly why logarithms are the inverse operation of exponentiation, and why they're so useful for working with quantities that grow or shrink exponentially - like earthquake magnitudes, sound decibels, and pH levels, all of which are logarithmic scales.

Common mistakes

Trying to find the log of zero or a negative number - logarithms are only defined for positive real numbers.
Confusing "log" (often meaning base 10 by default in many contexts) with "ln" (natural log, base e) - always check which base a formula or problem actually intends.

Frequently asked questions

What's the difference between log and ln?

"log" (without a specified base) conventionally means base 10 in most everyday and engineering contexts, while "ln" specifically means the natural logarithm, base e (approximately 2.71828) - a base that shows up naturally throughout calculus and continuous growth problems.

Why is e used as a base at all?

The number e arises naturally from continuous growth processes (like continuously compounding interest or population growth), and the natural logarithm has especially clean mathematical properties in calculus, which is why it appears so frequently in higher mathematics and science.

How does the change of base formula work?

It lets you calculate any base's logarithm using only natural log or common log (which calculators readily provide): log base b of x equals ln(x) divided by ln(b) - the same ratio regardless of which common base you use for the calculation.

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