Problem 61 - Intermediate Algebra

Question

Write each logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. See Example 6. $$ \log _{2} \frac{2 \sqrt[3]{x}}{y} $$

Step-by-Step Solution

Answer

\(1 + \frac{1}{3} \log_2 x - \log_2 y \)

Step 1: Identify the logarithmic components

We need to rewrite the given logarithm: \( \log_2 \left( \frac{2 \sqrt[3]{x}}{y} \right) \) into the sum and/or difference of logarithms.

Step 2: Apply the Quotient Rule of Logarithms

Using the quotient rule for logarithms, \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \), we can separate the logarithm into:\[ \log_2 \left( \frac{2 \sqrt[3]{x}}{y} \right) = \log_2 (2 \sqrt[3]{x}) - \log_2 y \]

Step 3: Apply the Product Rule of Logarithms

Next, for the term \( \log_2 (2 \sqrt[3]{x}) \), we use the product rule: \( \log_b (M \times N) = \log_b M + \log_b N \). Thus:\[ \log_2 (2 \sqrt[3]{x}) = \log_2 2 + \log_2 \sqrt[3]{x} \]

Step 4: Apply the Power Rule of Logarithms

For \( \log_2 \sqrt[3]{x} \), use the power rule: \( \log_b (x^r) = r \cdot \log_b x \). Here, \( \sqrt[3]{x} = x^{1/3} \), hence:\[ \log_2 \sqrt[3]{x} = \frac{1}{3} \log_2 x \]

Step 5: Combine and Simplify

Substitute the expression from Step 4 back into the equation from Step 3, and then combine it with the expression from Step 2:\[ \log_2 (2 \sqrt[3]{x}) = \log_2 2 + \frac{1}{3} \log_2 x \]Overall, we have:\[ \log_2 \left( \frac{2 \sqrt[3]{x}}{y} \right) = \log_2 2 + \frac{1}{3} \log_2 x - \log_2 y \]Since \( \log_2 2 = 1 \), the expression simplifies to:\[ 1 + \frac{1}{3} \log_2 x - \log_2 y \]

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a helpful tool when dealing with the division of two logarithmic expressions. In essence, it allows you to rewrite the log of a quotient as the difference of two separate logs. This is particularly useful because it simplifies complex expressions into more manageable components.

Here is how the rule is structured:

• \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)

In the given exercise, we applied this rule to split the logarithm of a fraction, \( \log_2 \left( \frac{2 \sqrt[3]{x}}{y} \right) \), into the difference of two individual logs: \( \log_2 (2 \sqrt[3]{x}) \) and \( \log_2 y \). This step helps to break down the expression into simpler parts that can be further simplified using other logarithmic rules, such as the product and power rules.

Once separated, each term can be handled individually, allowing for easier manipulation and simplification.

Product Rule of Logarithms

When dealing with the multiplication of two quantities within a logarithm, the product rule is your best friend. It simplifies a logarithm involving a product into the sum of two separate logarithms, making the expression easier to work with.

The product rule is as follows:

• \( \log_b (M \times N) = \log_b M + \log_b N \)

In our exercise, this rule was used to simplify \( \log_2 (2 \sqrt[3]{x}) \) into separate components: \( \log_2 2 \) and \( \log_2 \sqrt[3]{x} \).

By using the product rule, we can turn a more complicated expression into a simple addition of logs, which paves the way for further simplification using other logarithmic properties. This method is particularly useful for expressions where the terms contain one or more factors that could be individually expressed in logarithmic terms.

Power Rule of Logarithms

The power rule of logarithms is perfect for dealing with exponential expressions within a logarithm. This rule allows you to bring the exponent down in front of the log, transforming a complex exponential form into a multiplication, which is much easier to handle.

The power rule is defined by:

• \( \log_b (x^r) = r \cdot \log_b x \)

Applying this rule was crucial in simplifying the expression \( \log_2 \sqrt[3]{x} \). Recognizing that a cube root can be written as an exponent \((x^{1/3})\), we applied the power rule:

• \( \log_2 \sqrt[3]{x} = \frac{1}{3} \log_2 x \)

This simplification is key to rearranging the entire expression into a more straightforward form. By converting exponents into coefficients, the power rule enables the expression to be further simplified and combined with other logarithmic terms to achieve the final, simplified result.