Logarithms in simple words
A logarithm (log) answers one question: what power do I need? It is the opposite of an exponent.
Example: 10 × 10 = 100, so 10² = 100. That means the log base 10 of 100 is 2, written log₁₀(100) = 2. You need two 10s multiplied together to make 100.
Another example: 2 × 2 × 2 = 8, so log₂(8) = 3. The small number (the base) is what you multiply, and the answer tells you how many of them you need.
Two logs have short names. log with no base usually means base 10. ln (the natural log) means base e, a special number close to 2.718.
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.