Problem 40 - Precalculus

Question

Write each expression as a single logarithm. $$3 \log _{b} x-\log _{b} y$$

Step-by-Step Solution

Answer

\( \log_b \left( \frac{x^3}{y} \right) \)

Step 1: Apply the Power Rule

The power rule of logarithms states that \( a \log_b x = \log_b(x^a) \). Apply this rule to the term \( 3 \log_b x \) to rewrite it as \( \log_b(x^3) \).

Step 2: Use the Quotient Rule

The quotient rule of logarithms is \( \log_b a - \log_b y = \log_b\left(\frac{a}{y}\right) \). Apply this rule to combine the two logarithmic expressions \( \log_b(x^3) \) and \( \log_b y \).

Step 3: Combine Using the Quotient Rule

According to the quotient rule, combine \( \log_b(x^3) \) and \( \log_b y \) into a single logarithmic expression as follows: \( \log_b \left( \frac{x^3}{y} \right) \).

Key Concepts

Power Rule of Logarithms

The power rule of logarithms is a crucial concept that allows us to simplify expressions where a logarithm is multiplied by a constant. This rule can be mathematically expressed as: \( a \log_b x = \log_b(x^a) \). To put it simply, if you have a multiple of a logarithm, you can "move" the multiplier to become an exponent of the argument inside the logarithm. This transformation is useful in scenarios where you want to consolidate terms or simplify expressions.

For example, consider the expression \( 3 \log_b x \). Using the power rule, we convert it into a more simplified logarithm: \( \log_b(x^3) \). This approach is particularly helpful when dealing with logarithmic equations, as it allows for easier manipulation and understanding.

This rule gives us the ability to change the form of a logarithmic term, thus making complex problems more manageable. Whenever you are faced with a constant multiplying a logarithm, remember that turning that constant into an exponent can work wonders in simplifying your expression.

Quotient Rule of Logarithms

The quotient rule of logarithms helps us to connect subtraction within logarithmic expressions. According to this rule, if you have two logarithms with the same base being subtracted, you can write them as a single logarithm of a quotient. Mathematically, the rule is stated as: \( \log_b a - \log_b y = \log_b \left(\frac{a}{y}\right) \).

This functionality allows you to combine separate logarithms into a single expression. For instance, if we take \( \log_b(x^3) - \log_b y \), applying the quotient rule yields: \( \log_b \left( \frac{x^3}{y} \right) \). This simplifies the expression and reduces the number of logarithms to work with.

The key benefit of the quotient rule is to help solve and transform logarithmic equations, making them more straightforward. Recognizing where subtractive logarithm expressions occur will enable you to use this rule to combine into singular forms, which are often easier to evaluate or further manipulate.

Properties of Logarithms

The properties of logarithms form a fundamental toolkit for working with logarithmic equations and expressions. These properties include the power rule, the quotient rule, and the product rule, among others. Each of these helps in rewriting and simplifying logarithmic expressions.

To recap:

• **Power Rule:** Converts a constant multiplicator to an exponent, i.e., \( a \log_b x = \log_b(x^a) \).
• **Quotient Rule:** Transforms subtraction into division, i.e., \( \log_b a - \log_b y = \log_b \left(\frac{a}{y}\right) \).
• **Product Rule (not directly used in this problem):** Combines addition into multiplication, i.e., \( \log_b a + \log_b y = \log_b (a \cdot y) \).

Understanding these properties enables you to compress complex logarithmic expressions into simpler forms that are more intuitive and easy to solve.

Overall, by mastering these properties, you will be able to tackle a variety of logarithmic problems with confidence and clarity. They serve as the building blocks for advanced mathematical operations involving logarithms.