Logarithm Calculator

A logarithm answers the question: to what exponent must a given base be raised to produce a given number? Formally, log_b(x) = y means b^y = x. For example, log10(1000) = 3 because 10^3 = 1000.

ln (natural logarithm) uses base e ≈ 2.71828, favoured in calculus and sciences. log10 (common logarithm) uses base 10, used in pH, decibels, and the Richter scale. log2 (binary logarithm) uses base 2, standard in computer science and information theory.

log_b(x) = ln(x) / ln(b) = log10(x) / log10(b). This lets you compute a logarithm with any base using natural or common logarithms available on most calculators.

No — logarithms are only defined for positive real numbers (x > 0). For negative inputs or zero, the logarithm is undefined in the real number system. Complex logarithms exist but are outside the scope of this calculator.

log_b(1) = 0 for any valid base b. This is because b^0 = 1 by definition. The result is always zero regardless of the base you choose.

The base must be a positive real number (b > 0) and must not equal 1 (b ≠ 1). A base of 0 is undefined. A base of 1 is excluded because 1 raised to any power always equals 1, making it impossible to reach any other number.

Logarithms underlie many measurement scales: the Richter scale for earthquake magnitude, the decibel (dB) scale for sound intensity, pH for acidity, and star brightness in astronomy. They also govern compound interest calculations and population growth models.

log_b(x · y) = log_b(x) + log_b(y). This property means multiplying two numbers corresponds to adding their logarithms, which historically enabled rapid multiplication using logarithm tables.

log_b(x^n) = n · log_b(x). For example, log10(1000) = log10(10^3) = 3 · log10(10) = 3 · 1 = 3. The power rule is widely used in simplifying expressions in algebra and calculus.

Results are rounded to 8 decimal places using standard IEEE 754 double-precision floating-point arithmetic. For most practical applications this precision is more than sufficient. Extremely large or small inputs may be subject to rounding at the limits of floating-point representation.