Problem 86 - Algebra for College Students

Question

Express each of the following as a single logarithm. (Assume that all variables represent positive real numbers.) For example, $$ 3 \log _{b} x+5 \log _{b} y=\log _{b} x^{3} y^{5} $$ $$ \log _{b} x+\frac{1}{2} \log _{b} y $$

Step-by-Step Solution

Answer

\( \log_b (x \sqrt{y}) \)

Step 1: Understand Properties of Logarithms

Before we begin, remember the logarithmic properties: the multiple of a logarithm can be expressed as an exponent, i.e., \( a \cdot \log_b x = \log_b x^a \). Also, the sum of logarithms is the logarithm of the product, i.e., \( \log_b x + \log_b y = \log_b (xy) \).

Step 2: Apply the Power Rule

Use the power rule to rewrite each component separately. For the second term, apply the power rule: \( \frac{1}{2} \cdot \log_b y = \log_b y^{1/2} \). This gives us the expression \( \log_b x + \log_b y^{1/2} \).

Step 3: Apply the Product Rule

Now, apply the product rule of logarithms. The expression \( \log_b x + \log_b y^{1/2} \) becomes \( \log_b (x \cdot y^{1/2}) \).

Step 4: Simplify the Expression

Combine the expression into a single logarithm: \( \log_b (x \sqrt{y}) \). Here, \( y^{1/2} \) is equivalent to \( \sqrt{y} \).

Key Concepts

Logarithmic Identities

Logarithmic identities are fundamental tools in simplifying and manipulating logarithmic expressions. They allow us to express complex logarithmic statements in simpler forms, making calculations more manageable. Let’s go over some of the key identities:

• Product Rule: The logarithm of a product is the sum of the logarithms. Mathematically, this is represented as: \( \log_b (xy) = \log_b x + \log_b y \).
• Quotient Rule: The logarithm of a quotient is the difference between the logarithms. Given by: \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).
• Power Rule: The logarithm of a power is the exponent times the logarithm of the base. Notated as: \( \log_b x^a = a \log_b x \).

By understanding these basic logarithmic identities, you can transform almost any logarithmic expression into a form that is easier to work with or solve. These transformations are especially helpful in algebra and calculus for solving equations and simplifying expressions.

Power Rule

The power rule of logarithms is particularly useful when dealing with exponents within a logarithmic expression. It tells us how to "pull out" the exponent from the inside of the logarithm and multiply it outside. This rule is expressed as:\[ \log_b x^a = a \cdot \log_b x \]This transformation is incredibly useful in simplifying expressions where an exponent is involved. For instance, if you have an expression like \( 5 \cdot \log_b z \), you can rewrite it as \( \log_b z^5 \) using the power rule.

• This rule is applicable to any real number, allowing flexibility in simplifying a wide array of logarithmic expressions.
• It provides a method to manage exponents in logarithmic functions, making it easier to handle complex mathematical problems.

Understanding the power rule is vital for dealing with logarithms, as it helps in breaking down and simplifying logarithmic expressions in practical applications.

Product Rule

The product rule of logarithms helps in combining logarithms of products into a single expression. It states that the log of a product is equivalent to the sum of the logs of its individual factors. Mathematically, it is expressed as:\[ \log_b (xy) = \log_b x + \log_b y \]This rule is instrumental in simplifying logarithmic expressions by combining multiple logs into a singular, more manageable form. For example, if you're working with \( \log_b x + \log_b y \), using the product rule, you can simplify this to \( \log_b (xy) \).

• By employing the product rule, you can turn the sum of two separate logarithms into a single logarithm. This often makes it easier to evaluate or simplify an equation.
• The product rule emphasizes the additive nature of logarithms, showcasing how multiplication inside a log corresponds to addition outside.

Applying the product rule regularly makes tasks involving logs more efficient, allowing for quicker and cleaner algebraic manipulations in mathematics.