Logarithm rules

• Equivalence of logarithms and power notation:
  \begin{eqnarray*}
  a^{x}= y &\leftrightarrow & \log_{a} y = x \\
  e^{x}=y &\leftrightarrow& \ln y = x 
  \end{eqnarray*}
• Logarithm of a product rule:
  \[ \log_{a} (u\times v) = \log_{a}(u) +\log_{a}(v)\]
• Logarithm of a quotient or division rule:
  \[ \log_{a}\left( \frac{u}{v}\right) = \log_{a}(u) - \log_{a}(v)\]
• Logarithm of a power rule:
  \[ \log_a x^n = n \log_a x\]
• Logarithm of number to the same base \(\log_a a = 1\)
• Logarithm of \(1\) to any base: \(\log_a 1 = 0\)

Examples:

• \(\log_{3} 2 + \log_{3} x = \log_{3}(2\times x) = \log_{3} (2x)\)
• \(\log_{3} 2 - \log_{3} x = \log_{3} \left(\frac{2}{x}\right)\)
• Simplifying
  \begin{eqnarray*}
  3 \log 2 + \log 125&=& \log 2^3 + \log 125 \\
  &=& \log 8 + \log 125 \\
  &=& \log (8\times 125) \\
  &=& \log(1000) \\
  &=& 3
  \end{eqnarray*}
• Simplifying
  \begin{eqnarray*}
  \log xy^3 - \log x^2y + \log x
  &=& \log \left( \frac{xy^3}{x^2y}\times x \right) \\
  &=& \log y^2 \\
  &=& 2 \log y 
  \end{eqnarray*}