Problem 6 - Precalculus: Functions and Graphs

Question

Exer. 1-8: Express in terms of logarithms of \(x, y, z,\) or \(w\) $$\log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}}$$

Step-by-Step Solution

Answer

\( \frac{1}{2} \log y - 4 \log x - \frac{1}{3} \log z \).

Step 1: Simplify the Expression

The given expression is \( \log \frac{\sqrt{y}}{x^{4} \sqrt[3]{z}} \). Simplify this expression step by step before breaking it into known logarithmic forms.The square root \( \sqrt{y} \) can be expressed as \( y^{1/2} \) and the cube root \( \sqrt[3]{z} \) as \( z^{1/3} \). So the expression inside the logarithm becomes:\[ \frac{y^{1/2}}{x^4 z^{1/3}} \].

Step 2: Apply Logarithmic Properties

Use the properties of logarithms to simplify the expression.The property \( \log \frac{a}{b} = \log a - \log b \) helps here:\[ \log \frac{y^{1/2}}{x^4 z^{1/3}} = \log y^{1/2} - \log (x^4 z^{1/3}) \].Next, apply the property \( \log(ab) = \log a + \log b \):\[ \log y^{1/2} - (\log x^4 + \log z^{1/3}) \].

Step 3: Use Power Rule of Logarithms

Now apply the power rule: \( \log a^b = b \log a \).For \( \log y^{1/2} \), it becomes:\[ \frac{1}{2} \log y \].For \( \log x^4 \), it turns into:\[ 4 \log x \].For \( \log z^{1/3} \), it becomes:\[ \frac{1}{3} \log z \].

Step 4: Combine All Parts

Substitute the simplified logs back into the expression:\[ \frac{1}{2} \log y - (4 \log x + \frac{1}{3} \log z) \].Distribute the negative sign:\[ \frac{1}{2} \log y - 4 \log x - \frac{1}{3} \log z \].

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is fundamental when simplifying expressions. These properties help us manipulate logarithmic expressions efficiently. Here are some key properties we'll use:

• Quotient Rule: The rule states that for any positive numbers \(a\) and \(b\), with a base \(c\), \( \log_c \frac{a}{b} = \log_c a - \log_c b \). This property allows us to split a log of a quotient into a subtraction.
• Product Rule: This states that \( \log_c (ab) = \log_c a + \log_c b \), letting us express the log of a product as a sum.

In our solution, we first use the quotient rule to separate the numerator and denominator of the logarithmic expression. Then, for the denominator, we apply the product rule, turning it into a subtraction of two logs. This step is vital to make complex expressions more manageable.

Power Rule in Logarithms

The power rule is one of the most powerful tools in logarithms, allowing us to simplify expressions effectively. It states:

• If you have \( \log_c a^b \), you can rewrite it as \( b \log_c a \).

This rule is particularly useful when working with exponents. In the exercise, we apply the power rule to three different parts:

• \(\log y^{1/2}\) becomes \( \frac{1}{2} \log y \)
• \(\log x^4\) changes to \( 4 \log x \)
• \(\log z^{1/3}\) is simplified to \( \frac{1}{3} \log z \)

By doing this, we bring down the exponents and transform them into coefficients. This step significantly simplifies complex expressions and is critical for further simplification.

Logarithmic Simplification

Logarithmic simplification involves reducing expressions into their simplest form by applying the properties of logarithms. The goal is to make the expression easier to read and work with, especially in more advanced calculations.
In this problem, simplification begins by expressing roots as fractional exponents. For instance:

• The square root \(\sqrt{y}\) becomes \(y^{1/2}\)
• The cube root \(\sqrt[3]{z}\) becomes \(z^{1/3}\)

Then, we progressively apply logarithmic properties, starting with splitting the expression using quotient and product rules, followed by applying the power rule. Finally, combining all parts leads to a cleaner expression where each variable's contribution is explicit. This methodical approach aids in achieving a logarithmic expression in its simplest, most understandable form.

Fractional Exponents

Understanding fractional exponents is key to solving problems involving roots and powers more effectively.
When you see an expression like \(y^{1/2}\), it represents the square root of \(y\), and \(z^{1/3}\) represents the cube root of \(z\). These notations are crucial because they simplify the manipulation of expressions involving roots.

• Fractional Exponent Representation: The expression \(y^{1/n}\) corresponds to the \(n\)th root of \(y\).
• Conversion Between Forms: Converting from root form to fractional exponents makes it easier to use logarithmic properties, as they are directly applicable to exponents.

Recognizing and converting roots to fractional exponents helps streamline the problem-solving process, especially when dealing with logarithmic expressions. This understanding bridges the gap between roots and powers, ensuring you can apply fundamental rules seamlessly.