Log Calculator
Calculate the logarithm of a number with any base. Supports natural logarithm (ln), common logarithm (log10), and binary logarithm (log2).
Logarithm Base 10
Understanding Logarithms
A logarithm is the inverse operation to exponentiation. It answers the question: "To what power must we raise the base ($b$) to get the number ($x$)?"
logᵦ(x) = y means bʸ = x
- Base (b): The number being multiplied. Common bases are 10, 2, and $e$.
- Number (x): The value you want to reach. Must be positive.
- Result (y): The exponent.
Common Types of Logarithms
- Common Log (log): Uses base 10. Often written simply as $\log(x)$. Used in engineering (decibels) and science (pH scale).
- Natural Log (ln): Uses base $e$ (Euler's number ≈ 2.718). Written as $\ln(x)$. Fundamental in calculus and compound interest.
- Binary Log (log₂): Uses base 2. Essential in computer science and information theory.
Change of Base Formula
Most calculators only have buttons for $\ln$ and $\log_10$. To calculate a log with a different base (like $\log_5(100)$), we use the Change of Base formula:
logᵦ(x) = ln(x) / ln(b)
? Frequently Asked Questions
In the real number system, you cannot raise a positive base to any power and get a negative result (e.g., 10^y is always > 0). Therefore, the logarithm of a negative number is undefined.
Log(0) is undefined. As x approaches 0 from the positive side, the logarithm approaches negative infinity.
'Log' usually refers to base 10 (common logarithm), while 'ln' refers to base 'e' (natural logarithm). They behave similarly but are scaled differently.