Log Calculator
logb(x) = 0
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The Change-of-Base Formula
A logarithm logb(x) answers the question "b raised to what power equals x?" Calculators and programming languages generally only implement the natural logarithm (base e) directly, so any other base is computed using the change-of-base formula: logb(x) = ln(x) / ln(b), which is mathematically equivalent to logb(x) = log10(x) / log10(b). This calculator uses that identity, so the result is exact regardless of which base you enter — including non-integer bases.
Why Both x and the Base Must Be Positive
Logarithms are only defined for positive real arguments (x > 0), and the base must be positive and not equal to 1 (b > 0, b ≠ 1) — a base of 1 would make b raised to any power always equal 1, so it could never reach most values of x. If you enter values outside these ranges, the calculator will flag the input as invalid rather than silently showing a wrong answer. This is different from an exponent calculator, which reverses the question by finding bx instead of solving for the exponent.
Logarithms vs. Roots and Scientific Notation
Logarithms are closely tied to exponential growth and decay — for example, the half-life calculator relies on natural logs to solve for decay time. They're also useful for reading very large or very small numbers: the common logarithm (base 10) of a number roughly tells you its order of magnitude, which is the same idea behind scientific notation.
Frequently Asked Questions
How do you calculate a logarithm with a base your calculator doesn't have a button for?
Use the change-of-base formula: log_b(x) = ln(x) / ln(b), where ln is the natural logarithm. Since virtually every calculator and programming language provides ln (or log base 10), you can compute a logarithm in any base by dividing the natural log of x by the natural log of the base. This calculator applies that formula automatically for whatever base you enter.
Why can't the base be 1, and why must x be positive?
The base must be positive and not equal to 1 because 1 raised to any power always equals 1, so log base 1 could never solve for most values of x (and base 0 or negative bases produce undefined or non-real results for most exponents). Similarly, x must be greater than 0 because no real exponent applied to a positive base can ever produce a zero or negative result.