A logarithm is the inverse operation of exponentiation. If b^y = x, then log_b(x) = y. In other words, the logarithm answers the question: "To what power must the base be raised to get this number?"
Logarithms are used extensively in science, engineering, and mathematics. They help simplify complex calculations, measure quantities that span many orders of magnitude (like pH, decibels, and earthquake magnitudes), and solve exponential equations.
The most common logarithms are:
To calculate a logarithm with any base, use the change of base formula:
Key Properties:
Problem: Calculate log₁₀(1000)
Solution:
Problem: Calculate ln(e²)
Solution:
Problem: Calculate log₂(8)
Solution:
"log" typically refers to base 10 logarithm (common logarithm), while "ln" refers to base e logarithm (natural logarithm). The natural logarithm uses Euler's number e ≈ 2.71828 as its base.
In real numbers, no. The logarithm is only defined for positive real numbers. For negative numbers, you need to use complex numbers and the complex logarithm.
The natural logarithm arises naturally in calculus and many natural phenomena. The derivative of ln(x) is simply 1/x, and it appears in many growth/decay processes, making it "natural" for mathematical descriptions of nature.
log(0) is undefined. No matter what power you raise a positive base to, you can never get 0. The logarithm function approaches negative infinity as x approaches 0 from the positive side.