Properties of Logarithms

Fundamental Concept
Logarithmic functions are the inverse of exponential functions: if ( y = b^x ), then ( x = log_b(y) ).

Key Properties of Logarithms

  • Logarithm of 1:

    • ( log_b(1) = 0 )

    • Explanation: Any base raised to the zero power equals one.

  • Logarithm of the base:

    • ( log_b(b) = 1 )

    • Explanation: Any base raised to the first power equals the base itself.

Inverse Property

  • The inverse property states: ( log_b(b^x) = x \ (x > 0) )

  • Example:

    • To evaluate ( log_{10}(10^2) ): use the inverse property to obtain ( 2 ).

One-to-One Property

  • The one-to-one property states: ( logb(M) = logb(N) ) if and only if ( M = N ).

  • Usage:

    • If ( log3(3x) = log3(2x + 5) ), then set the arguments equal: ( 3x = 2x + 5 ).

Product Rule for Logarithms

  • Logarithm of a product: ( logb(MN) = logb(M) + log_b(N) \ (b > 0) )

  • Derivation using inverse properties.

  • Example:

    • ( log3(30x(3x+4)) = log3(30) + log3(x) + log3(3x + 4) ).

Quotient Rule for Logarithms

  • Logarithm of a quotient: ( logb(M/N) = logb(M) - log_b(N) \ (b > 0) )

  • Example:

    • For ( log2(15x(x-