Log Calculator
Logarithm & Change-of-Base Tool
Understanding Logarithms
A logarithm is the inverse of exponentiation — instead of asking "what do I get when I raise b to the power n?", it asks "what power do I need to raise b to, to get x?"
If b^n = x, then log_b(x) = n. Logarithms let us "undo" exponential growth and turn multiplicative relationships into additive ones — the reason slide rules and logarithmic scales work at all.
Base 10 (the common log) is the everyday choice for scales like pH and the Richter scale. Base e (the natural log) shows up naturally in calculus and any process — like compound interest or radioactive decay — that grows or shrinks continuously.
The Change-of-Base Formula
Since most devices only compute natural (or common) logarithms directly, this identity lets you find a logarithm in any base by dividing two natural logs — exactly the work this calculator shows step by step.
Where Logarithms Show Up
Logarithms appear throughout science and finance: the Richter scale for earthquake magnitude, the decibel scale for sound intensity, pH for acidity, and the time-to-double formulas used in compound interest all rely on logarithmic math to compress enormous ranges of values into manageable numbers.
Used for pH, decibels, and the Richter scale.
Euler's number, ≈ 2.71828, central to calculus and continuous growth.
Logarithms are only defined for positive arguments.
Key Takeaways
- A logarithm answers "what power?" — it's the inverse operation of raising a base to an exponent.
- The change-of-base formula lets you compute a logarithm in any base using only natural (or common) logs.
- The argument must be positive, and the base must be positive and not equal to 1, or the logarithm is mathematically undefined.
Frequently Asked Questions
- Enter the value
xyou want the logarithm of. - Enter the base
b, or click "Common (log₁₀)" or "Natural (ln)" to instantly fill in base 10 or base e. - Click "Calculate" to see the result along with the change-of-base work.
The logarithm log_b(x) answers the question "to what power must b be raised to get x?" For example, log₂(8) = 3 because 2³ = 8. Logarithms are the inverse operation of exponentiation.
A common log (written log(x)) always uses base 10, and is common in scientific measurement scales like pH and decibels. A natural log (written ln(x)) uses the mathematical constant e ≈ 2.71828 as its base, and appears constantly in calculus, compound growth, and decay problems.
Most calculators only have built-in buttons for base-10 and base-e logarithms, so to find a logarithm with any other base, we use the change-of-base formula: log_b(x) = ln(x) / ln(b). This calculator shows that exact division so you can see how the final answer is derived.
Logarithms are only defined for positive arguments, because no real power of a positive base can ever produce zero or a negative number. Entering zero or a negative value for x will show a clear error instead of an incorrect result.
A base of 1 raised to any power always equals 1, so log base 1 could never uniquely solve for x — the function is undefined there. Bases of 0 or negative numbers also break down for the same reason, since they can't reliably produce every positive real number as a power. The calculator blocks base values of 1 and anything ≤ 0.