Definition: LogarithmicFunction

Definition: Logarithmic Function

A logarithmic function is of the form \(\log_b(x)\) where \(b>0\) and \(b \neq 1\).

Note: A logarithmic function is not an algebraic function.

Key Characteristics of Logarithmic Function

• Domain is positive real numbers
• Range is all real numbers
• When \(b > 1\) the function is increasing on its entire domain.
  • As \(x\) gets close to zero from the right the output approaches negative infinity.
  • As \(x\) increases the output increases without bound.
  • These functions are concave down.
• When \(0< b < 1\) the function is decreasing on its entire domain.
  • As \(x\) gets close to zero from the right the output grows without bound (approaches infinity).
  • As \(x\) increases the output approaches negative infinity.
  • These functions are concave up.

Properties of Logarithms

If \(a>0\), \(b > 0\), \(c > 0\), \(b \neq 1\), and \(n\) is any real number, then - \(\log_b b^x = x\) and \(b^{\log_b x} = x\) (inverse property) - Special Cases - \(\log_b b = 1\) - \(\log_b 1 = \log_b b^0 = 1\) - \(\log_b(ac) = \log_b(a) + \log_b(c)\) (product property) - \(\log_b\left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)\) (quotient property) - \(\log_b(a^n) = n\log_b(a)\) (power property)