Logarithm Calculator
Compute log_b(x) for any base — common log (base 10), natural log (base e), or any custom base. Includes change-of-base details.
What is Logarithm Calculator?
The logarithm calculator answers the question: "What power do I raise the base to in order to get x?" For example, log₁₀(1000) = 3 because 10³ = 1000.
It computes log of any positive argument in any positive base ≠ 1 — including the most common bases used in U.S. math classes: 10 (common log), e (natural log), and 2 (binary log).
Formula
By definition: logb(x) = y means by = x.
Change-of-base formula (used internally for any base):
logb(x) = log(x) ÷ log(b) = ln(x) ÷ ln(b)
The argument x must be positive. The base must be positive and not equal to 1.
Worked example
log₁₀(1000) = 3 (because 10³ = 1000).
ln(e) = 1 (definition of natural log).
log₂(64) = 6 (because 2⁶ = 64).
How to use this calculator
- Enter the argument x (must be positive).
- Enter the base — 10 for common log, 2.71828 (or e) for natural log, 2 for binary log, or any custom positive value not equal to 1.
- The result for your chosen base appears, plus the most common bases (10, e, 2) for reference.
Frequently asked questions
What's the difference between log and ln?
On a U.S. calculator, log means log base 10 (common log) and ln means log base e (natural log, where e ≈ 2.71828). Outside the U.S., "log" sometimes means natural log — context matters.
Why must the argument be positive?
Because by > 0 for any real y when b > 0. There is no real exponent that makes a positive base equal a negative number, so log of a negative is undefined in real math.
What's a real-world use of logarithms?
Decibels (sound), pH (chemistry), Richter scale (earthquakes), and stellar magnitude (astronomy) are all logarithmic scales. Each compresses a huge range of values into a manageable scale.
How do logarithms appear in algebra class?
Solving exponential equations: 2x = 50 means x = log₂(50) ≈ 5.64. Calculator-required for non-perfect powers.