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The power rule for logarithms is one of the fundamental properties that help simplify logarithmic expressions. This rule states that if you have a logarithm of a number raised to a power, you can bring that power down in front as a multiplier. In mathematical terms, it's expressed as:

• For a given base, \(\log_b(a^n) = n \cdot \log_b(a)\).

Let's break this down with an example from our exercise: consider the expression \(\log (x/y)^{1/5}\).
The entire fraction \(x/y\) is raised to the power of \(1/5\).
Using the power rule, we simplify it by taking the exponent \(1/5\) in front of the logarithm.
So, it becomes \(\frac{1}{5} \cdot \log(x/y)\).
This transformation is helpful because it reduces the complexity of the logarithmic expression, making it easier to work with further.

The quotient rule is essential in handling logarithms that involve division.
This rule simplifies a logarithm of a quotient into a subtraction problem, separating the log of the numerator and the log of the denominator.
In formulaic terms, it is described as:

• For the logarithm of a quotient, \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\).

Using this property in our exercise, we apply the quotient rule to \(\frac{1}{5} \cdot \log(x/y)\).
The expression inside the logarithm function \(x/y\) splits into \(\log x - \log y\) due to the quotient property.
Thus, now the expression is \(\frac{1}{5} \cdot (\log x - \log y)\), which is a more expanded form and often makes evaluating or further manipulation much simpler.
This step is crucial to breaking down complex logs into simpler, smaller parts that can be easily tackled.

Expanding logarithmic expressions involves utilizing the properties of logarithms to rewrite the expression in a more detailed form.
In general, expansion helps in breaking down a compounded logarithmic expression into its individual logarithmic components.
The expansion process makes use of several rules, such as the power rule and the quotient rule, to achieve this.
Let's align this with our example: starting with \(\log \sqrt[5]{\frac{x}{y}}\), an expression containing a root and a quotient.

• First, we identify that the root is \(1/5\) and bring it out in front of the logarithm using the power rule: \(\frac{1}{5} \cdot \log(x/y)\).
• Then, we utilize the quotient rule to further expand into \(\frac{1}{5}(\log x - \log y)\).

Each step systematically unpacks the expression, maximizing simplicity and clarity.
Consequently, expanded expressions are often easier to differentiate, integrate, or plug different values into, as needed for various calculations.