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The power rule of logarithms helps to transform logarithmic equations where coefficients are involved. This rule states that if you have a logarithm written as \( a \log_b(M) \), you can effectively bring the coefficient \( a \) into the argument as an exponent, creating \( \log_b(M^a) \). This is extremely helpful when working to simplify or combine logarithmic expressions.

In our example, the expression \( \frac{1}{2} \log_3(x) \) can be transformed using the power rule. We bring the coefficient \( \frac{1}{2} \) as the exponent of \( x \) inside the logarithm, creating \( \log_3(x^{1/2}) \). Similarly, for \( -2 \log_3(y) \), the coefficient \(-2\) becomes the exponent of \( y \) (i.e., \( y^{-2} \)), resulting in \( \log_3(y^{-2}) \). The third part, \( -\log_3(z) \), translates to \( \log_3(z^{-1}) \) by considering -1 as the exponent.

This transformation is the first step in simplifying expressions because it allows you to handle all logarithmic terms with uniformity, easing further manipulation.

Combining logarithms is based on the use of the properties of logarithms, which allow you to condense expressions. The rule \( \log_b(M) - \log_b(N) = \log_b\left(\frac{M}{N}\right) \) is particularly useful for combining terms.

In the original exercise, after applying the power rule, we have three separate terms: \( \log_3(x^{1/2}) \), \( \log_3(y^{-2}) \), and \( \log_3(z^{-1}) \). We can combine these by using the properties of logarithms. Start with the first two terms, combining them using the property above:

\[ \log_3(x^{1/2}) + \log_3(y^{-2}) = \log_3\left(x^{1/2} \cdot y^{-2}\right) \]

Then include the third term, \( \log_3(z^{-1}) \):

\[ \log_3(x^{1/2} \cdot y^{-2}) + \log_3(z^{-1}) = \log_3\left(x^{1/2} \cdot y^{-2} \cdot z^{-1}\right) \]

Now, all terms are combined into a single logarithm expression, which is more concise and easier to handle.

Once logarithms are combined into a single expression, as illustrated in this exercise, it enables you to tidy up and interpret the logarithm more cleanly.

The outcome of combining the terms is \( \log_3(x^{1/2} \cdot y^{-2} \cdot z^{-1}) \). This expression combines all previous individual logarithms into one, achieved by organizing the terms inside the argument into a product. We apply exponent rules here to clarify:

- A root \( x^{1/2} \) can be expressed as \( \sqrt{x} \).
- A negative exponent \( y^{-2} \) translates to \( \frac{1}{y^2} \).
- Likewise, \( z^{-1} \) becomes \( \frac{1}{z} \).

Thus, putting it all together, we organize into a fraction within a single logarithm:

\[ \log_3\left(\frac{\sqrt{x}}{y^2z}\right) \]

This simplification is often the final goal in algebraic expressions involving logarithms, as it optimizes comprehension and further calculations required for solving equations. You can now see the relationships between \( x \), \( y \), and \( z \) more clearly, making this expression more insightful and useful for analysis or decision-making in subsequent steps.