Formula and method
Logarithms reverse exponentiation. In bʸ = x, y = ln(x)/ln(b), x = bʸ and b = exp(ln(x)/y). Real logarithms require x > 0 and b > 0, b ≠ 1. A base-solving problem with y = 0 does not have a unique base.
Worked example
For log₁₀(100), ln(100)/ln(10) = 2 because 10² = 100. For log₂(8), the answer is 3. ln(e) = 1.
Inputs and limitations
A base between 0 and 1 is valid and reverses the direction of growth: log₀.₅(4) = −2. A negative logarithm does not mean its argument is negative. Floating-point overflow and underflow can prevent a finite answer.