Log Calculator

Calculate logarithms in any base — log₁₀, ln, log₂, and custom base.

Logarithm Calculator

Calculate logarithms in any base — log₁₀, ln (natural log), log₂, and custom base.

Calculation Inputs

Number (x)

0.00110,000,000

Logarithm Base

21,000

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

Calculation Results

log₍10₎(100)

log₁₀ (Common)

ln (Natural)

4.605

log₂ (Binary)

6.644

Loading Smart Logic...

Quick Tip

Enter the number you want the logarithm of. It must be greater than zero.

About This Tool

Why use Log Calculator?

A logarithm answers the question "what power do I need?". log₁₀(1000) = 3 because 10³ = 1000. This calculator returns the logarithm in any base you choose, alongside the three most used bases: log₁₀ (common), ln (natural, base e ≈ 2.71828) and log₂ (binary).

Logs turn multiplication into addition, which is why they appear wherever quantities span huge ranges: decibels, pH, the Richter scale, and the complexity of algorithms. Only positive numbers have real logarithms, and the base must be positive and not equal to 1.

Key Benefits

Any base, plus the three standard ones

Custom bases use the change-of-base rule, and log₁₀, ln and log₂ are always shown for comparison.

Clear errors instead of silent nonsense

Zero, negative numbers and a base of 1 have no logarithm, and the tool says so rather than showing a misleading value.

Useful for growth and scale problems

Logs answer "how many times does this double" and "how many digits does this number have".

How to use Log Calculator

1
Enter the number you want the logarithm of. It must be greater than zero.
2
Choose the base — 10 for common logs, e for natural logs, 2 for binary, or any other positive value.
3
Read the result in your chosen base, plus the standard log₁₀, ln and log₂ values.
4
Check the answer by raising the base to the result: it should return your original number.

Best use cases

Solve exponential equations in algebra homework.

Work out how many years an investment takes to double.

Convert between linear values and decibel or pH scales.

Estimate the depth of a binary search or tree in computer science.

Log Calculator FAQs

What is the difference between log and ln?

Written without a base, log usually means base 10 in engineering and school maths. ln is the natural logarithm, base e ≈ 2.71828, which is the one that appears in growth and decay formulas and in calculus.

What is the change-of-base formula?

log_b(x) = ln(x) ÷ ln(b), or equally log₁₀(x) ÷ log₁₀(b). That is how this calculator handles custom bases, and how you can compute any base on a calculator that only has log and ln keys.

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

Raising a positive base to any real power always gives a positive result, so no exponent produces zero or a negative number. As the input approaches zero, the logarithm falls towards negative infinity.