Logarithmic Properties


Change-of-Base Formula:
Let \(a,b,\) and \(x\) be positive real numbers such that \(a \not = 1\) and \(b \not = 1.\) Then \(\log_a x\) can be converted to a different base as follows

  • Base \(b\):
    $$
    \log_a x = \frac{\log_b x}{\log_b a}
    $$
  • Base \(10\):
    $$
    \log_a x = \frac{\log x}{\log a}
    $$
  • Base \(e\):
    $$
    \log_a x = \frac{\ln x}{\ln a}
    $$
Properties of Logarithms:
Let \(a\) be a positive number \((a \not = 1)\), \(n\) be a real number, and \(u\) and \(v\) be positive numbers.

  • Product Property:
    $$
    \begin{aligned}
    \log_a(uv) & = \log_a u + \log_a v \\
    \ln(uv) & = \ln u + \ln v
    \end{aligned}
    $$
  • Quotient Property:
    $$
    \begin{aligned}
    \log_a \left( \frac{u}{v} \right) & = \log_a u – \log_a v\\
    \ln \left( \frac{u}{v} \right) & = \ln u – \ln v
    \end{aligned}
    $$
  • Power Property:
    $$
    \begin{aligned}
    \log_a u^n &= n \log_a u \\
    \ln u^n &= n \ln u
    \end{aligned}
    $$
  • $$
    \log_a a = 1
    $$
  • $$
    \log_a 1 = 0
    $$