Calculate logarithms with any custom base, natural log ln(x), common log log10(x), binary log log2(x), and change of base steps.
Use natural logarithms to evaluate logs with arbitrary bases.
Calculate the natural log of numerator and denominator.
Therefore, 10^3.0000 = 1000.
A logarithm is the inverse function of exponentiation. The logarithm of a number x to a base b (written as log_b(x)) answers the question: "To what exponent must we raise the base b in order to equal x?" Logarithms measure physical scales spanning immense magnitudes, such as sound decibels (dB), earthquake Richter scales, and chemical pH acidity.
The log of a product equals the sum of the logs: log_b(x · y) = log_b(x) + log_b(y).
The log of a quotient equals the difference of the logs: log_b(x / y) = log_b(x) - log_b(y).
The log of a power pulls the exponent out as a multiplier: log_b(xᵏ) = k · log_b(x).
log_b(b) = 1 for any valid base, and log_b(1) = 0 because b⁰ = 1.
Evaluate log₂(64) exactly, and approximate log₅(100) using the change of base formula.
Because any positive base raised to any real power always produces a strictly positive result (bʸ > 0). No real exponent y can make bʸ equal 0 or a negative number.
The natural logarithm is the logarithm to the base e (Euler number ≈ 2.718281828459). It arises naturally in calculus, compound interest, physics growth/decay models, and probability.
Because 1 raised to any power is always 1 (1ʸ = 1). An equation like 1ʸ = 5 has no solution, and 1ʸ = 1 has infinitely many solutions, so base 1 cannot define a function.
Human ear sensitivity is logarithmic. Sound pressure level in decibels is calculated as dB = 20 · log₁₀(P / P₀), where P is sound wave pressure and P₀ is the threshold of hearing.
Understanding your calculation
A logarithm answers which exponent produces a value: log base b of x equals y when b^y = x.
log base 10 of 1000 is 3 because 10³ = 1000; the natural logarithm uses base e.
For real logarithms, x must be positive and the base must be positive and different from one. Logarithms of sums cannot be split into sums of logarithms.