91Ó°ÊÓ

Understanding the rules of logarithms is essential for solving logarithmic equations effectively. These rules are mathematical properties that allow us to manipulate logarithmic expressions in various ways to simplify them or make them solvable. Key rules include:

• The Product Rule, which states that the log of a product is equal to the sum of the logs of its factors, written as \( \text{log}_b(mn) = \text{log}_b(m) + \text{log}_b(n) \).
• The Power Rule, which allows us to bring the exponent in a logarithm out front, turning \( \text{log}_b(m^n) = n \times \text{log}_b(m) \).
• The Quotient Rule, which says that the log of a quotient is the difference between the logs of the numerator and denominator, expressed as \( \text{log}_b(\frac{m}{n}) = \text{log}_b(m) - \text{log}_b(n) \).
• The Change of Base Formula allows to convert a log from one base to another, useful when dealing with less common log bases.

These logarithm rules are interconnected and can often be applied in sequence to simplify complex logarithmic expressions.

The quotient rule is particularly useful when dealing with logarithmic expressions that involve division. This rule transforms the log of a quotient into a subtraction problem, thereby simplifying the original expression. The formal expression of the quotient rule is:
\( \text{log}_b(\frac{m}{n}) = \text{log}_b(m) - \text{log}_b(n) \)
This rule is instrumental when simplifying the logarithm of a fraction. For example, in our exercise, we apply the quotient rule to the expression \( \frac{1}{2} \text{log}_{4} (\frac{x}{y}) \), effectively breaking down the log of a fraction into the difference of the logs of its numerator and denominator before multiplying by the fractional exponent.

Rational exponents represent roots and allow us to write expressions involving roots as exponents. This is based on the principle that \( m^{1/n} \) is the nth root of m. A rational exponent, such as \( 1/2 \), signifies a square root when in the denominator. In the step-by-step solution, we converted the square root in \( \text{log}_4 \( \sqrt{\frac{x}{y}} \) \) into a rational exponent. The expression then reads \( \text{log}_4 (\frac{x}{y})^{1/2} \). This conversion makes it possible to apply the power rule of logarithms, bringing the exponent to the front and greatly simplifying the equation for further manipulation.

Simplifying logarithmic expressions involves applying the logarithm rules systematically. In our textbook exercise, simplification is achieved through a stepwise approach. After converting to rational exponents, we use the power and quotient rules to split the expression. This results in \( \frac{1}{2} \text{log}_{4} x - \frac{1}{2} \text{log}_{4} y \), a much more straightforward form. This process not only makes the evaluation of logs easier but also lays the groundwork for solving complex logarithmic equations. Simplification is vital for both understanding the underlying mathematics and for practical computation, especially when dealing with logs in different bases or those with complicated arguments.