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Logarithms - How to Calculate Them ?

(What's a Logarithm?)

Given a logarithm whose base is a fraction, such as: \[log_{\frac{1}{2}}\begin{pmatrix}8\end{pmatrix}\] we can calculate/evaulate this using the following formula.

Logarithm with a Fractional Base

Given a positive whole number \(b\) and \(x>0\), the following will always be true: \[log_{\frac{1}{b}}\begin{pmatrix}x\end{pmatrix} = -log_b\begin{pmatrix}x\end{pmatrix}\]

Tutorial

In the following tutorial we explain the formula for logarithms with a fraction as a base.

Exercise

Calculate each of the following without a calculator:

  1. \(log_{\frac{1}{2}}\begin{pmatrix}4\end{pmatrix}\)
  2. \(log_{0.5}\begin{pmatrix} 8 \end{pmatrix}\)
  3. \(log_{\frac{1}{3}}\begin{pmatrix}9\end{pmatrix}\)
  4. \(log_{0.2}\begin{pmatrix}125\end{pmatrix}\)
  5. \(log_{\frac{1}{4}}\begin{pmatrix}64\end{pmatrix}\)
  6. \(log_{\frac{1}{5}}\begin{pmatrix}625\end{pmatrix}\)
  7. \(log_4\begin{pmatrix}64\end{pmatrix} + log_{\frac{1}{3}}\begin{pmatrix}27\end{pmatrix}\)
  8. \(log_5\begin{pmatrix}125\end{pmatrix} - log_{\frac{1}{4}}\begin{pmatrix}16\end{pmatrix}\)
  9. \(log_{\frac{1}{7}}\begin{pmatrix}49\end{pmatrix} + log_{0.1}\begin{pmatrix}1000\end{pmatrix} + 2\)
  10. \(log_{0.2}\begin{pmatrix}25\end{pmatrix} + log_{\frac{1}{4}}\begin{pmatrix}256\end{pmatrix} + log_2\begin{pmatrix}512\end{pmatrix}\)

Note: this exercise can be downloaded as a worksheet to practice with: worksheet

Answers Without Working

  1. \(log_{\frac{1}{2}}\begin{pmatrix}4\end{pmatrix} = -2\)
  2. \(log_{0.5}\begin{pmatrix} 8 \end{pmatrix} = -3\)
  3. \(log_{\frac{1}{3}}\begin{pmatrix}9\end{pmatrix} = -2\)
  4. \(log_{0.2}\begin{pmatrix}125\end{pmatrix} = -3\)
  5. \(log_{\frac{1}{4}}\begin{pmatrix}64\end{pmatrix} = -3\)
  6. \(log_{\frac{1}{5}}\begin{pmatrix}625\end{pmatrix} = -4\)
  7. \(log_4\begin{pmatrix}64\end{pmatrix} + log_{\frac{1}{3}}\begin{pmatrix}27\end{pmatrix} = 0\)
  8. \(log_5\begin{pmatrix}125\end{pmatrix} - log_{\frac{1}{4}}\begin{pmatrix}16\end{pmatrix} = 5\)
  9. \(log_{\frac{1}{7}}\begin{pmatrix}49\end{pmatrix} + log_{0.1}\begin{pmatrix}1000\end{pmatrix} + 2 = -3\)
  10. \(log_{0.2}\begin{pmatrix}25\end{pmatrix} + log_{\frac{1}{4}}\begin{pmatrix}256\end{pmatrix} + log_2\begin{pmatrix}512\end{pmatrix} = 3\)


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