Problem 68 - 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. $$ \log _{b} \sqrt[4]{\frac{x^{3} y^{2}}{z^{4}}} $$

Step-by-Step Solution

Answer

\( \log_b(\sqrt[4]{\frac{x^3 y^2}{z^4}}) \) simplifies to \( \frac{3}{4}\log_b(x) + \frac{1}{2}\log_b(y) - \log_b(z) \).

Step 1: Apply the Root and Quotient Properties

Start by applying the fourth root as an exponent of 1/4 to each term in the logarithm. Also, apply the quotient rule: \( \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \).\[ \log_b \left( \sqrt[4]{\frac{x^3 y^2}{z^4}} \right) = \log_b \left( \left( \frac{x^3 y^2}{z^4} \right)^{\frac{1}{4}} \right) \]Using the power property of logarithms, this becomes:\[ \frac{1}{4} \log_b \left( \frac{x^3 y^2}{z^4} \right) \]Now apply the quotient rule:\[ \frac{1}{4} \left( \log_b(x^3 y^2) - \log_b(z^4) \right) \]

Step 2: Apply the Product and Power Properties

Next, apply the product rule \( \log_b(MN) = \log_b(M) + \log_b(N) \) and the power rule \( \log_b(M^n) = n \log_b(M) \).For \( \log_b(x^3 y^2) \):\[ \log_b(x^3 y^2) = \log_b(x^3) + \log_b(y^2) \]Now, apply the power property:\[ \log_b(x^3) = 3\log_b(x) \]\[ \log_b(y^2) = 2\log_b(y) \]So:\[ \log_b(x^3 y^2) = 3\log_b(x) + 2\log_b(y) \]Apply the power rule to \( \log_b(z^4) \):\[ \log_b(z^4) = 4\log_b(z) \]

Step 3: Combine all Terms

Plug in the simplified expressions from the previous step back into the expression:\[ \frac{1}{4} \left( 3 \log_b(x) + 2 \log_b(y) - 4 \log_b(z) \right) \]Distribute the \( \frac{1}{4} \) across the terms:\[ \frac{3}{4}\log_b(x) + \frac{1}{2}\log_b(y) - \log_b(z) \]

Step 4: Final Solution

The expression has been rewritten as the sum and/or difference of logarithms of single quantities.The final simplified expression is:\[ \frac{3}{4}\log_b(x) + \frac{1}{2}\log_b(y) - \log_b(z) \]

Key Concepts

Quotient Rule

The quotient rule for logarithms is a handy tool in math. It helps in breaking down complex logarithmic expressions. The rule states:

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

This rule is applicable when you have a fraction inside a logarithm. By using the quotient rule, you can split this fraction into separate parts.
For our example, applying the quotient rule allows us to express:

• \( \log_b\left( \frac{x^3 y^2}{z^4} \right) \)
• as \( \log_b(x^3 y^2) - \log_b(z^4) \)

This step simplifies the computation by separating it into two simpler logarithmic expressions. It's a powerful technique to simplify fractions inside logarithms.

Product Rule

The product rule for logarithms deals with multiplying terms inside a logarithm. This rule states:

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

You use this rule when you encounter a product inside a logarithmic expression. The rule helps convert the product into a sum of two logarithms.
In our original problem, for the term:

• \( \log_b(x^3 y^2) \)

we apply the product rule to express it as:

• \( \log_b(x^3) + \log_b(y^2) \)

This simplifies the task by focusing on each factor inside the logarithm separately. Applying the product rule is a crucial step in breaking down the expression further.

Power Rule

The power rule is used when an exponent is involved in the logarithmic expression. According to this rule:

• \( \log_b(M^n) = n \log_b(M) \)

The power rule allows you to take an exponent out of the logarithm, turning it into a multiplier. This can make complex equations much easier to manage.
For example, in our exercise, it was used on terms like:

• \( \log_b(x^3) = 3 \log_b(x) \)

By applying the power rule, each instance of an exponent can be expressed as a product of the exponent and the logarithm. Similarly, for \( \log_b(z^4) \) it becomes:

• \( 4 \log_b(z) \)

This transformation simplifies multiplying terms and allows for easy integration with other rules like the quotient and product rules. It provides a pathway to simplify and evaluate these expressions effectively.