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The power rule is an essential property of logarithms that simplifies expressions where numbers multiply the logarithm. In mathematics, the power rule is expressed as \[\log_b{a^n} = n \log_b{a}\]Here, \(b\) is the base, \(a\) is the argument, and \(n\) represents the power or exponent. This rule allows us to move the coefficient in front of the logarithm inside, transforming it into an exponent. For example, if you have \(-2 \log y\), the power rule enables you to rewrite it as \(\log y^{-2}\). This transformation is particularly useful when combining multiple logarithm terms into a single logarithmic expression. Make sure to identify coefficients and always apply this rule first when simplifying logarithmic expressions.

The product rule is another fundamental property of logarithms that lets us combine two logarithmic expressions. This rule is stated as: \[\log_b{(a \cdot c)} = \log_b{a} + \log_b{c}\]With this rule, if you have a logarithm of a product, you can split it into a sum of individual logarithms. For example, if you have \(\log x + \log{z^3}\), you can combine these two into \(\log (x \cdot z^3)\). Likewise, if you see a sum of logs and need to condense them, recognize that each term can represent a factor of a product. This rule is pivotal when compressing multiple logarithm terms into a single log expression, simplifying calculations and making expressions compact.

The subtraction rule is closely related to the product rule, but it involves dividing, instead of multiplying. This rule is expressed as: \[\log_b{\left( \frac{a}{c} \right)} = \log_b{a} - \log_b{c}\]When you have one logarithm subtracted from another, this rule allows you to rewrite it as a single logarithm of a fraction. For instance, in the expression \(\log x - \log{y^2}\), the subtraction indicates division, leading to \(\log \left( \frac{x}{y^2} \right)\). Therefore, subtraction turns the argument of the subtracted log into the denominator of a fraction. This is a valuable tool in simplifying logs since division often arises when manipulating or solving logarithmic expressions.

Condensing logarithms involves using logarithm rules to transform multiple logarithmic expressions into a single one. This technique makes the expressions more concise and easier to work with. To condense an expression like \[\log x - 2 \log y + 3 \log z\]Follow the steps:

• **Apply the Power Rule**: Move coefficients by converting them to exponents, changing the expression to: \(\log x - \log y^2 + \log z^3\).
• **Use the Product Rule**: Identify terms being added, \(\log x + \log z^3\), and combine them to: \(\log (x \cdot z^3)\).
• **Apply the Subtraction Rule**: Notice the subtraction, \(- \log y^2\), which implies division. Hence, it modifies the expression into: \(\log \left( \frac{x \cdot z^3}{y^2} \right)\).

These steps highlight the combination of rules needed to condense a logarithmic expression efficiently, turning complex terms into a singular, streamlined form.