Summary and Main Ideas

Summary

• The logarithm of a number \((x)\) with a certain base \((a)\) is equal to the exponent \((y)\), the value to which that certain base must be raised to equal the number \((x)\).

  If \(x = {a}^{y}\), then \(y = {\log}_{a}(x)\), where \(a>0\), \(a \ne 1\) and \(x>0\).

• Logarithms and exponentials are inverses of each other.

  \(f(x) = \log_{a}{x} \quad\) and \(\quad f^{-1}(x) = a^{x}\)

• Common logarithm: \(\log{a}\) means \(\log_{\text{10}}{a}\)
• The “LOG” function on your calculator uses a base of \(\text{10}\).
• Natural logarithm: \(\ln\) uses a base of \(e\).

• Special values:

  • \(a^{0} = 1 \qquad \log_{a}{1} = 0\)
  • \(a^{1} = a \qquad \log_{a}{a} = 1\)

• Logarithmic laws:

  • \(\log_{a}{xy} = \log_{a}{x} + \log_{a}{y} \qquad (x > 0 \text{ and } y > 0)\)
  • \(\log_{a}{\cfrac{x}{y}} = \log_{a}{x} - \log_{a}{y} \qquad (x > 0 \text{ and } y > 0)\)
  • \(\log_{a}{x^{b}} = b \log_{a}{x} \qquad (x > 0)\)
  • \(\log_{a}{x} = \cfrac{\log_{b}{x}}{\log_{b}{a}} \qquad (b > 0 \text{ and } b \ne 1)\)

• Special reciprocal applications:

  • \(\log_{a}{x} = \cfrac{1}{\log_{x}{a}}\)
  • \(\log_{a}{\cfrac{1}{x}} = - \log_{a}{x}\)