Problem 79 - Algebra for College Students

Question

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume that all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b}\left(x \sqrt{\frac{x}{y}}\right) $$

Step-by-Step Solution

Answer

\( \frac{3}{2} \log_b x - \frac{1}{2} \log_b y \)

Step 1: Identify the logarithmic expression

The given logarithmic expression is \( \log_b \left( x \sqrt{\frac{x}{y}} \right) \). We need to express it as a sum or difference of simpler logarithmic quantities.

Step 2: Apply the property of logarithms for products

We use the property \( \log_b (mn) = \log_b m + \log_b n \). Here, \( m = x \) and \( n = \sqrt{\frac{x}{y}} \), so we write: \[ \log_b \left( x \sqrt{\frac{x}{y}} \right) = \log_b x + \log_b \sqrt{\frac{x}{y}}. \]

Step 3: Express the square root as a power

The expression \( \sqrt{\frac{x}{y}} \) can be rewritten as \( \left(\frac{x}{y}\right)^{1/2} \). Substitute this back into the logarithmic expression: \[ \log_b \sqrt{\frac{x}{y}} = \log_b \left(\frac{x}{y}\right)^{1/2}. \]

Step 4: Use the power rule for logarithms

The power rule states \( \log_b (m^n) = n \log_b m \). Apply this rule: \[ \log_b \left(\frac{x}{y}\right)^{1/2} = \frac{1}{2} \log_b \left(\frac{x}{y}\right). \]

Step 5: Simplify using the quotient rule for logarithms

The quotient rule states \( \log_b \frac{m}{n} = \log_b m - \log_b n \). Apply this rule: \[ \frac{1}{2} \log_b \left(\frac{x}{y}\right) = \frac{1}{2} (\log_b x - \log_b y). \]

Step 6: Combine the results

Substitute back into the expression from Step 2: \[ \log_b x + \log_b \sqrt{\frac{x}{y}} \] becomes \[ \log_b x + \frac{1}{2} (\log_b x - \log_b y). \] Distribute the \( \frac{1}{2} \): \[ \log_b x + \frac{1}{2} \log_b x - \frac{1}{2} \log_b y. \] Then, simplify by combining like terms: \[ \frac{3}{2} \log_b x - \frac{1}{2} \log_b y. \]

Key Concepts

Logarithm Properties

Logarithm properties are essential to understand for dealing with complex logarithmic expressions. These properties include the product rule, the quotient rule, and the power rule.

• The **product rule** states that the logarithm of a product is equal to the sum of the logarithms of its factors: \( \log_b(mn) = \log_b m + \log_b n \).
• The **quotient rule** provides that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator: \( \log_b \frac{m}{n} = \log_b m - \log_b n \).
• The **power rule** tells us that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number: \( \log_b (m^n) = n \log_b m \).

Applying these rules allows us to break down complex logarithmic expressions into simpler components, making them easier to manipulate and understand. By using these properties, we can express difficult quantities in terms of more straightforward logarithms.

Logarithmic Simplification

Logarithmic simplification involves rewriting logarithmic expressions to make them easier to work with. This is often accomplished by using logarithmic properties to transform complex expressions into simpler forms. For example, consider the expression \( \log_b \left( x \sqrt{\frac{x}{y}} \right) \). By applying the product rule, we can separate this into \( \log_b x + \log_b \sqrt{\frac{x}{y}} \), breaking it down into more manageable components.Further simplification can be achieved by expressing square roots as fractional exponents, such as \( \sqrt{\frac{x}{y}} = \left(\frac{x}{y}\right)^{1/2} \). Applying the power rule here, we find: \( \log_b \left(\frac{x}{y}\right)^{1/2} = \frac{1}{2} \log_b \left(\frac{x}{y}\right) \).Finally, applying the quotient rule helps simplify \( \frac{1}{2} \log_b \left(\frac{x}{y}\right) \) to \( \frac{1}{2} (\log_b x - \log_b y) \), which further reduces the complexity of the expression. Using these steps consistently allows for efficient simplification.

Logarithmic Rules

Logarithmic rules are the foundational guidelines that allow us to manipulate and understand logarithmic expressions effectively. These rules arise naturally from the properties of exponents and provide a consistent system for working with logarithms.When applying the logarithm rules, it's crucial to remember:

• The **base** of the logarithm should remain consistent throughout your calculations. Changing the base requires specific transformations, like the change of base formula.
• The **order** of operations matters; be attentive to the use of parentheses to ensure accurate simplification.
• Always keep variables within the logarithmic expressions positive real numbers, as logarithms of non-positive numbers are not defined in the realm of real numbers.

With these rules in mind, describing complex expressions like \( \log_b \left( x \sqrt{\frac{x}{y}} \right) \) becomes accessible by applying sequential logarithmic rules, transforming it into a more straightforward form such as \( \frac{3}{2} \log_b x - \frac{1}{2} \log_b y \). This highlights the power and utility of logarithmic rules in simplifying mathematical expressions.