Problem 31 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \ln \left(x \sqrt{\frac{y}{z}}\right) $$

Step-by-Step Solution

Answer

\( \ln x + \frac{1}{2} \ln y - \frac{1}{2} \ln z \)

Step 1: Identify the Inner Components

The expression given is \( \ln \left(x \sqrt{\frac{y}{z}}\right) \). Identify the inner components that will be affected by the logarithmic laws. Here, the expression inside the natural logarithm is \( x \cdot \sqrt{\frac{y}{z}} \).

Step 2: Apply the Product Rule of Logarithms

The Product Rule states that \( \ln(a \cdot b) = \ln a + \ln b \). Apply this rule to break down the logarithm of the product inside: \( \ln \left(x \cdot \sqrt{\frac{y}{z}}\right) = \ln x + \ln \left(\sqrt{\frac{y}{z}}\right) \).

Step 3: Use the Power Rule on the Square Root

The square root can be expressed as a power: \( \sqrt{\frac{y}{z}} = \left(\frac{y}{z}\right)^{1/2} \). The Power Rule states that \( \ln(a^b) = b \ln a \). Apply this to get: \( \ln \left(\sqrt{\frac{y}{z}}\right) = \frac{1}{2} \ln \left(\frac{y}{z}\right) \).

Step 4: Apply the Quotient Rule

The Quotient Rule states that \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \). Apply this to expand the term within the logarithm: \( \ln\left(\frac{y}{z}\right) = \ln y - \ln z \).

Step 5: Combine the Results

Substitute back the expansion from Step 3 into the expression from Step 2: \( \ln x + \frac{1}{2} (\ln y - \ln z) \). This simplifies to: \( \ln x + \frac{1}{2} \ln y - \frac{1}{2} \ln z \).

Key Concepts

Product Rule of Logarithms

When you encounter a logarithm of a product, the Product Rule of Logarithms becomes useful. This rule helps in breaking down complex expressions into simpler parts, making them easier to work with.
The Product Rule states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, it is expressed as:
\( \ln(a \cdot b) = \ln a + \ln b \).

• This rule is often used to separate multiple terms inside a logarithm.
• It simplifies the process of expanding expressions by breaking them into smaller, more manageable parts.

In our example, we applied the Product Rule to the expression \( \ln \left(x \cdot \sqrt{\frac{y}{z}}\right) \), which resulted in breaking it down to \( \ln x + \ln \left(\sqrt{\frac{y}{z}}\right) \). This helps separate the terms, setting the stage for further simplification with other logarithmic rules.

Power Rule of Logarithms

The Power Rule of Logarithms is another powerful tool that simplifies expressions involving exponents. This rule allows you to bring the exponent in front of the logarithm, turning complex powers into simple multipliers.
The Power Rule can be written as follows:
\( \ln(a^b) = b \cdot \ln a \).

• This rule is particularly handy when dealing with powers or roots, like square roots or cube roots.
• Converting exponential expressions inside logarithms into products makes calculations straightforward.

In the exercise, the square root \( \sqrt{\frac{y}{z}} \) was initially identified and rewritten using the power of \( 1/2 \) as \( \left(\frac{y}{z}\right)^{1/2} \). Applying the Power Rule, it was simplified to \( \frac{1}{2} \ln \left(\frac{y}{z}\right) \). This unraveling is a crucial step in very complex expansions.

Quotient Rule of Logarithms

When faced with a logarithm of a quotient, the Quotient Rule of Logarithms is the guideline to follow. This rule decomposes a quotient within a logarithm into a difference of logarithms, providing clarity and simplifying further manipulation.
The Quotient Rule is stated as:
\( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).

• The rule effectively separates division within logarithms, simplifying expressions that include fractions.
• It eliminates the complexity of handling division inside a single logarithm, spreading it into two subtractive terms.

In the given problem, this rule allowed us to expand \( \ln \left(\frac{y}{z}\right) \) as \( \ln y - \ln z \). This step is vital in breaking down the expression into linear components that can be easily handled in further calculations.