🪵Logarithm Calculator
Find log base b of x for any positive base and value.
Free Logarithm Calculator
This logarithm calculator finds log_b(x) — "log base b of x" — for any positive base b ≠ 1 and positive value x, using the change-of-base formula: log_b(x) = ln(x) / ln(b). This works because most calculators and programming languages only provide natural log (ln) or common log (log₁₀) directly.
Use base 10 for the common logarithm, or base e (≈2.71828) for the natural logarithm — quick preset buttons are provided for both. Any other base, like log₂ for binary problems, works the same way.
Free, instant, and runs entirely in your browser — no sign-up required.
About the Logarithm Calculator
Features
Any base
Compute logarithms in base 2, 10, e, or any other positive base.
Common presets
One-click buttons for base 10 and base e.
Working shown
See the change-of-base formula applied and a result check.
Instant, private
Runs entirely in your browser. No data is sent anywhere.
Frequently Asked Questions
What does log_b(x) mean?
log_b(x) answers the question: 'to what power must b be raised to get x?' For example, log₂(8) = 3 because 2³ = 8.
What is the change-of-base formula?
log_b(x) = ln(x) / ln(b) — it lets you compute a logarithm in any base using only natural log (or common log), which is what calculators and programming languages provide directly.
What's the difference between log and ln?
'log' (without a subscript) usually means base-10 (common) logarithm, while 'ln' specifically means the natural logarithm, base e (≈2.71828).
Why can't the base be 1?
1 raised to any power always equals 1, so log base 1 could never produce any value other than 1 — the function is undefined for base 1.
Why must the value be positive?
No real power of a positive base ever produces zero or a negative number, so logarithms of zero or negative numbers are undefined in the real number system.
What is e, and why is it used?
e (≈2.71828) is a fundamental mathematical constant that appears naturally in growth, decay, and calculus. The natural logarithm (base e) is the standard logarithm used throughout higher mathematics.
How do I verify a logarithm result?
Raise the base to the power of the computed result — it should equal your original value. The calculator shows this check automatically.
Can the base be a fraction?
Yes, any positive base other than 1 works, including fractions like 0.5 — the change-of-base formula handles it the same way.
Where are logarithms used?
Logarithms appear in pH calculations, decibel scales, the Richter scale, compound interest and growth models, computer science (binary logs), and solving exponential equations.
Is this calculator free to use?
Yes, completely free with no sign-up, and it runs entirely in your browser.