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Logarithms have special rules that simplify their expressions. One of these rules is the power rule. It is extremely useful when you are dealing with coefficients in logarithms. The power rule states that if you have a logarithm in the form of \(a \log b\), it can be rewritten as \(\log(b^a)\).

This means that the coefficient in front of the logarithm can become an exponent for the term inside the logarithm. For example, in the expression \(2 \log x\), the number 2 can be moved to become an exponent of \(x\).

Thus, \(2 \log x \) can be simplified to \( \log(x^2)\). This trick allows you to prepare complex logarithmic terms for further simplification using other rules, such as the quotient rule.

The quotient rule is another key rule when working with logarithms. It helps when there is a subtraction of two logs with the same base. The quotient rule states: \( \log a - \log b = \log\left( \frac{a}{b}\right) \).

When you see a subtraction of logs, you can combine them into one log by dividing their terms. For example, if you have \(\log(x^2) - \log y\), you can rewrite it as \( \log\left(\frac{x^2}{y}\right)\).

This application of the quotient rule takes advantage of the fact that division and subtraction are linked in logarithmic rules, similar to multiplication and addition. The quotient rule simplifies your expressions into a single log, which is often the desired form.

Simplifying logarithmic expressions involves using different rules to create a cleaner, more compact logarithmic form. This process can make calculations easier and expressions easier to understand.

In our exercise, we've used the power rule to convert \(2 \log x\) into \( \log(x^2)\). Then, we applied the quotient rule to combine \( \log(x^2) - \log y\) into \( \log\left(\frac{x^2}{y}\right) \).

These transformations reduce the number of terms and help everything fit into a single logarithm. The benefit is that this form is often what other mathematical operations require, particularly in calculus and algebra. Simplifying expressions is an essential skill when dealing with more complex equations, where clarity and simplicity are key.