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Understanding the power rule of logarithms is essential when simplifying logarithmic expressions. This rule enables us to rewrite a logarithm that has a coefficient in front of it. The rule states that if you have a coefficient in front of a log term, such as in 2 log\textsubscript{3} u, you can move the coefficient as an exponent to the argument of the log. Mathematically, this is written as:

• \t$$a \log_b(x) = \log_b(x^a)$$

Applying the power rule to our example, we transform 2 log\textsubscript{3} u into log\textsubscript{3}(u\(u^2\)). This allows us to proceed with further simplification. Remember, moving coefficients to become exponents can make your expressions simpler and help in combining them later on.

In logarithmic expressions, you often need to combine or simplify logs. One helpful rule is the subtraction rule of logarithms. This rule allows you to combine two logarithms that are being subtracted. The rule is as follows:

• \t$$\log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)$$

After applying the power rule, we are left with log\textsubscript{3}(u\(u^2\)) - log\textsubscript{3}(v). To combine these, we use the subtraction rule. This results in a single log expression. So, log\textsubscript{3}(u\(u^2\)) - log\textsubscript{3}(v) becomes log\textsubscript{3}\(\left( \frac{u^2}{v}\right)\).
This rule is highly effective when simplifying complex logarithmic expressions. It condenses two logarithms into one, making the expression more manageable.

Combining multiple logarithms into a single expression is a common task. Once we've applied the power rule and the subtraction rule, we can finalize the expression into a single logarithm. To recap, we started with 2 log\textsubscript{3} u - log\textsubscript{3} v.
First, we used the power rule to rewrite it as log\textsubscript{3}(u\(u^2\)). Next, applying the subtraction rule, we combined it into log\textsubscript{3}\(\frac{u^2}{v}\).
This results in a simplified, single logarithm expression:

• $$\log_3\left(\frac{u^2}{v}\right)$$

Creating a single logarithm expression often makes it easier to work with and understand in equations. This step-by-step approach ensures clarity and enforces understanding of these fundamental logarithmic rules.