Andromeda
Note

Laws of Logarithms

Definition

The Laws of Logarithms are properties that allow for the manipulation and simplification of logarithmic expressions. They are the direct result of the properties of exponents, as logarithms are exponents.

Why It Matters

Logarithms turn products into sums—they are the “shortcuts” of the mathematical world. Mastering these laws is essential for handling exponential growth, signal processing, and any system where the scale of change is as important as the change itself.

Core Concepts

Let a>0,a≠1,A>0,B>0a > 0, a \neq 1, A > 0, B > 0, and CC be any real number.

  • Product Law: log⁡a(AB)=log⁡aA+log⁡aB\log_a(AB) = \log_a A + \log_a B

    • How to read: “The log base a of the quantity A times B equals the log base a of A plus the log base a of B.”
    • Meaning: Multiplying inside the log becomes adding outside—turns products into sums.

  • Quotient Law: log⁡a(AB)=log⁡aA−log⁡aB\log_a\left(\frac{A}{B}\right) = \log_a A - \log_a B

    • How to read: “The log base a of the fraction A over B equals the log base a of A minus the log base a of B.”
    • Meaning: Division inside the log becomes subtraction outside.

  • Power Law: log⁡a(AC)=Clog⁡aA\log_a(A^C) = C \log_a A

    • How to read: “The log base a of A to the power C equals C times the log base a of A.”
    • Meaning: Exponents come down as multipliers—essential for solving exponential equations.

  • Change of Base Formula: log⁡bx=log⁡axlog⁡ab\log_b x = \frac{\log_a x}{\log_a b}

    • How to read: “The log base b of x equals the ratio of log base a of x to log base a of b.”
    • Meaning / when to use: Evaluate any-base logs using ln or log on a calculator.
    • Most common versions: log⁡bx=log⁡xlog⁡b=ln⁡xln⁡b\log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}

Connected Concepts

Connected notes