Calculate the logarithm of a number to any base instantly.
logbase(x)
3.00
ln(x)6.907755
log₁₀(x)3.00
log₂(x)9.965784
Logarithm answers: "What power do I raise the base to in order to get the value?" For example, log₁₀(100) = 2 because 10² = 100.
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What is the Log Calculator?
A logarithm answers the question: 'What power must I raise the base to in order to get the value?' For example, log₁₀(100) = 2 because 10² = 100. Logarithms reverse exponentiation and are essential in mathematics, science, engineering, and data analysis.
How it works
Enter a base (2, 10, e, or any positive number except 1) and a value greater than zero. The calculator uses the change-of-base formula: log_base(x) = ln(x) / ln(base). It instantly displays the result and also shows ln(x), log₁₀(x), and log₂(x) for reference.
The change-of-base formula lets you compute any logarithm using natural logarithms or base-10 logarithms. Simply divide the logarithm of the value by the logarithm of the base, both in the same system.
Examples
Input
Result
Notes
Base 10, Value 100
2
log₁₀(100) = 2 because 10² = 100
Base 2, Value 32
5
log₂(32) = 5 because 2⁵ = 32
Base e, Value 7.389
2
ln(7.389) ≈ 2 because e² ≈ 7.389
How to use the Log Calculator
Enter the base in the top field (e.g. 10, 2, e, or any positive number except 1)
Or click a quick-base button: Base 10 (log), Base e (ln), or Base 2
Enter the value (x) in the bottom field — must be greater than 0
Read your result instantly in the results box
View ln(x), log₁₀(x), log₂(x) in the stats panel for additional insight
Benefits
Instant results — no manual computation or calculator fumbling
Any base supported — use common bases (2, 10, e) or any custom base
Change-of-base formula built-in — reference your work visually
Extra stats displayed — see natural log and common logs side-by-side
Perfect for STEM students and professionals — essential for exponential and logarithmic problems
Free, no sign-up — use on any device instantly
Tips & common mistakes
Common mistakes
Entering a base of 0 or 1 — logarithms with base 1 are undefined
Using a negative base — bases must be positive
Entering zero or negative values for x — logarithms only exist for positive x
Confusing log₁₀ with ln — log₁₀ uses base 10; ln uses base e ≈ 2.718
Forgetting that log_base(1) always equals 0 — because any base to the power 0 is 1
Tips
Remember: log_base(base) = 1 — the base raised to the power 1 is itself
Use ln for natural exponential decay (bacteria, radioactive half-lives); use log₁₀ for pH, decibels, and Richter scales
Logarithmic scales compress large ranges into smaller visuals — essential for graphs with huge value spreads
On a scientific calculator, ln is the natural log button; log is base-10 (though some label them differently)
Frequently asked questions
What's the difference between log and ln?
Log usually means log₁₀ (base 10), while ln is the natural logarithm (base e ≈ 2.718). Both answer 'what power' but to different bases. In computer science and math, ln is more fundamental.
Why can't the base be 1?
Because 1 to any power is always 1. So log₁(x) would be undefined or infinite for most x. Bases must be positive and not equal to 1.
Why must x be positive?
Logarithms of zero and negative numbers are undefined in real mathematics. You cannot raise a positive base to any real power and get a negative number or zero.
What is the change-of-base formula?
It lets you convert any logarithm: log_a(x) = log_b(x) / log_b(a). Use this to compute logs with bases not on your calculator.
How do I use logarithms in real life?
Logs model exponential growth (populations, investments), measure sound (decibels), acidity (pH), earthquakes (Richter), and data compression. Any 'doubling time' or 'half-life' problem involves logs.
Is my data private?
Yes. This is a client-side tool—all calculations happen in your browser. Your numbers are never sent to a server.