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The change of base formula is a very useful tool when working with logarithms. It allows you to rewrite a logarithm in terms of bases that are more familiar or easier to handle, like base 10 (common logarithm) or base e (natural logarithm). To use the formula, you can express a logarithm of any base as a fraction of two base 10 or natural logarithms. Specifically, the formula looks like this:

\(\text{log}_a(b) = \frac{\text{log}(b)}{\text{log}(a)}\text{or} \frac{\text{ln}(b)}{\text{ln}(a)}\)

This is extremely handy when your calculator only has buttons for base 10 (log) and base e (ln) logarithms. For instance, if you need to find \(\text{log}_{1/5}(27)\), you'd change it to \(\frac{\text{log}(27)}{\text{log}(1/5)}\). The formula simplifies complicated logarithms to types your calculator can handle directly.

Once you have rewritten your logarithm using the change of base formula, you can easily use a calculator to find the value. Most calculators have buttons for \(\text{log}\) (base 10) and \(\text{ln}\) (base e) functions. Let's go through the steps:

• First, input the value of \(\text{log}\) or \(\text{ln}\) that needs to be calculated. For example, type \(\text{log}(27)\).
• Press the equal button to get the result. Suppose this gives you approximately 1.4314.
• Clear your calculator, then type in the next value, which is \(\text{log}(1/5)\).
• Press the equal button again to get your second result. This should be about -0.69897.

After obtaining these values, the next step is to divide the first result by the second to find \(\text{log}_{1/5}(27)\). This completes the process and gives you the final answer, approximately -2.0488.

Understanding logarithm properties can simplify solving problems significantly. Here are a few key properties:

• Product Rule: \(\text{log}_a(b \times c) = \text{log}_a(b) + \text{log}_a(c)\)
• Quotient Rule: \(\text{log}_a(\frac{b}{c}) = \text{log}_a(b) - \text{log}_a(c)\)
• Power Rule: \(\text{log}_a(b^c) = c \times \text{log}_a(b)\)
• Change of Base Formula: \(\text{log}_a(b) = \frac{\text{log}_c(b)}{\text{log}_c(a)}\)
• Zero Rule: \(\text{log}_a(1) = 0\)
• Identity Rule: \(\text{log}_a(a) = 1\)

These properties help in rearranging and simplifying logarithmic expressions. For instance, in the given problem, we utilized the Change of Base Formula to manage a difficult logarithm by converting it to base 10, making it easy to calculate using a standard calculator.