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Logarithm Calculator: Log, Ln and Any Base

This free logarithm calculator finds the log of a number in any base. It also shows the natural log, common log and binary log together, plus the antilogarithm.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

A logarithm answers the question: what power gives this number? Logs turn multiplication into addition, which is why they power slide rules, decibels, pH, earthquake scales and information theory. Most calculators only have log10 and ln keys, so other bases take extra steps.

Enter the number and a base. Leave the base as e to get the natural log. The tool also lists the three most common logs at once.

How to Use This Tool

  1. Enter x, which must be greater than zero.
  2. Enter the base, for example 10, 2 or e.
  3. Press Calculate.
  4. Read log base b, and the ln, log10 and log2 rows.

How the Calculation Works

Using the change of base formula, log_b(x) = ln(x) ÷ ln(b). The antilogarithm, b^x, undoes the log. The base must be positive and not equal to 1.

Worked Example

log base 10 of 1,000 is 3. log base 2 of 64 is 6. ln of 20 is 2.9957. log base 5 of 125 is 3, because 5³ = 125.

Common Uses

  • Solving exponential equations.
  • Measuring pH, decibels and the Richter scale.
  • Counting how many bits a number needs.
  • Compound interest and doubling time.

Tips and Common Mistakes

  • log(a × b) = log a + log b, and log(a ÷ b) = log a − log b.
  • Type e for the base to use the natural log.
  • log base b of b is always 1.

Next, you may need the exponent and root calculator, scientific notation converter, quadratic equation solver, percentage calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

The logarithm of x to base b is the power you must raise b to in order to get x. Since 10³ = 1000, log base 10 of 1000 is 3.
It is the logarithm with base e, about 2.71828. It is written ln x and is used in growth, decay and calculus.
No power of a positive base gives zero or a negative result, so the real logarithm is undefined there.
log base b of x = ln x ÷ ln b. It lets you compute any base from the natural log, which is what this tool does.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.