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Introduction to Logarithms

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*}