Problem 124 - College Algebra

Question

In Exercises \(121-124\), determine whether each statement makes sense or does not make sense, and explain your reasoning. I expanded \(\log _{4} \sqrt{\frac{x}{y}}\) by writing the radical using a rational exponent and then applying the quotient rule, obtaining \(\frac{1}{2} \log _{4} x-\log _{4} y\)

Step-by-Step Solution

Answer

Yes, the expanded expression of \(\log _{4} \sqrt{\frac{x}{y}}\) is indeed \(\frac{1}{2} \log _{4} x - \frac{1}{2} \log _{4} y\). The logarithmic rules were correctly applied.

Step 1: Convert Radical to Rational Exponent

Rewrite the square root using a rational exponent. Remember that the square root of any quantity is equivalent to raising that quantity to the power of 1/2. Therefore, \(\sqrt{\frac{x}{y}}\) can be written as \(\left(\frac{x}{y}\right)^{1/2}\). So now, the expression becomes \(\log_4 \left(\frac{x}{y}\right)^{1/2}\).

Step 2: Apply Power Rule of Logarithms

The power rule states that the exponent of the argument inside the logarithm can be brought out as a coefficient of the logarithm, i.e., \(log_b(m^n) = n \cdot log_b(m)\). Applying this rule, our expression becomes \(1/2 \cdot \log_4 \frac{x}{y}\).

Step 3: Apply Quotient Rule of Logarithms

Now, for the final step, we can apply the quotient rule, i.e., \(log_b (a/c) = log_b (a) - log_b (c)\). Applying this rule to \(\frac{1}{2}\cdot log_4 (\frac{x}{y})\), we get \(\frac{1}{2} \times \log_4 x - \frac{1}{2} \times log_4 y\), which simplifies to \(\frac{1}{2} \log_4 x - \frac{1}{2} \log_4 y\).

Key Concepts

Rational Exponents

Rational exponents are a way to express roots as powers. They are particularly useful when you need to simplify expressions involving roots or when using logarithms. Essentially, a rational exponent is written as a fraction. The most common instance is the square root, which is represented by an exponent of \(\frac{1}{2}\). For instance,

• \( \sqrt{a} \) can be written as \( a^{1/2} \).
• Likewise, \( \sqrt[3]{a} \) can be rewritten as \( a^{1/3} \).

These expressions allow for easier manipulation in equations and logarithmic functions.
In the given exercise, the square root \( \sqrt{\frac{x}{y}} \) is transformed into \( \left(\frac{x}{y}\right)^{1/2} \). This conversion is crucial for further simplification using logarithmic rules.

Power Rule of Logarithms

The power rule of logarithms is a tool that helps simplify logarithmic expressions when the argument of the log is raised to a power. According to this rule,

• \( \log_b(m^n) = n \cdot \log_b(m) \)

This means that any exponent in the argument of the logarithm can be brought out as a coefficient in front of the log. In our exercise, the expression \( \log_4 \left(\frac{x}{y}\right)^{1/2} \) makes perfect use of this rule by turning it into \( \frac{1}{2} \cdot \log_4 \frac{x}{y} \).
This approach simplifies the expression and prepares it for further operations, such as the application of the quotient rule of logarithms.

Quotient Rule of Logarithms

The quotient rule of logarithms is a key principle for dealing with division inside a logarithmic argument. This rule states that:

• \( \log_b(\frac{a}{c}) = \log_b(a) - \log_b(c) \)

The rule essentially allows you to separate the divider from the dividend, simplifying the expression by splitting the logarithm into two parts. In our case, after initially applying the power rule, the quota is then tackled with the quotient rule.
The expression \( \frac{1}{2} \cdot \log_4 \frac{x}{y} \) splits into \( \frac{1}{2} \log_4 x - \frac{1}{2} \log_4 y \).
This step significantly simplifies the logarithmic expression by breaking it into simpler, more manageable parts.

Expression Simplification

Simplifying logarithmic expressions is akin to organizing a messy room. You want it to be neat and malfunction-free for better use. Throughout the solution process, several steps are employed to gradually break down and simplify a complex logarithmic expression.

• First, we converted a square root into a rational exponent.
• Second, we applied key logarithmic rules like the power and quotient rules.

Every step in this process is necessary to achieve a final, simpler expression.
These rules not only make the expression more straightforward but also clearer, allowing for easier interpretation or further mathematical manipulations.
In the end, simplifying expressions is a powerful technique in mathematics, aiding in both computation and comprehension.