Log Calculator
Calculate log base b of x, including common log (base 10) and natural log (base e).
- Input Value (x)100
- log_10(x)2
Logarithms compress scale — 100 collapses down to just 2 once you ask "what power of 10 gives this?"
- Natural Log ln(x)4.60517
ƒShow your work
- 1A logarithm asks "what power do I raise the base to?" — log_10(100) = 2
- 2Check: 10^2 ≈ 100
About the Log Calculator
A logarithm answers the question 'what power do I need to raise this base to, to get this number' — it's the inverse operation of an exponent, the same way a root is, just structured differently: a root solves for the base, a logarithm solves for the exponent. It's a genuinely strange concept the first time it's taught, and it shows up again later in contexts that don't look related at first glance — the pH scale, decibels, earthquake magnitude, and computer science's use of log base 2 to describe how efficiently an algorithm scales.
This calculator computes log base b of any value x, for any valid base, plus the natural log (base e) alongside it, since natural log is what most higher math and science actually uses once you're past an intro course.
How it’s calculated
log_b(x) asks 'b raised to what power equals x?' log₁₀(1000) = 3 because 10³ = 1000. This calculator computes it using the change-of-base formula: log_b(x) = ln(x) / ln(b), which works for any base by routing the calculation through natural log.
The natural log, written ln(x), is just log base e, where e (≈2.71828) is a special constant that shows up naturally in continuous growth and decay — compound interest compounded continuously, radioactive decay, population growth models. It's the default 'log' used throughout calculus and most of higher mathematics.
Logarithms are only defined for positive values, and only for a positive base that isn't 1 — you can't raise 1 to any power and get anything other than 1, so log base 1 would have no consistent answer.
Frequently asked questions
What's the difference between log and ln?
'log' with no base written usually means base 10 (common log) in everyday math, while 'ln' always specifically means base e, the natural log. They're the same operation with different bases — this calculator can compute either by setting the base to 10 or to e (≈2.71828).
Why can't you take the log of zero or a negative number?
Because no power of a positive base can ever produce zero or a negative number — a positive base raised to any real exponent always stays positive. Since a logarithm is asking 'what exponent gives this result,' there's no valid answer when the target isn't positive.
What is a logarithm used for in real life?
Anywhere a scale needs to be compressed because the underlying values span many orders of magnitude — the Richter scale for earthquakes, the decibel scale for sound, the pH scale for acidity, and computer science's use of log base 2 to describe how an algorithm's run time scales with input size.
How do you switch between different log bases?
Using the change-of-base formula: log_b(x) = log_k(x) / log_k(b) for any convenient base k, usually 10 or e since those are what calculators natively support. This is exactly how this calculator computes a log in any base you enter.
Related calculators
Powered by GetCalculator.online