Problem 3 - Algebra A Combined Function

Question

Write each sum as a single logarithm. Assume that variables represent positive numbers. $$ \log _{4} 9+\log _{4} x $$

Step-by-Step Solution

Answer

\(\log_4 (9x)\)

Step 1: Understand the Properties of Logarithms

The sum of two logarithms with the same base can be combined into a single logarithm using the Product Property of Logarithms. This property states that \( \log_b a + \log_b c = \log_b (a \cdot c) \).

Step 2: Identify the Terms

We have the logarithmic terms \( \log_4 9 \) and \( \log_4 x \). According to the Product Property, these can be combined because they both have the same base, which is 4.

Step 3: Apply the Product Property

Using the Product Property of Logarithms, combine \( \log_4 9 \) and \( \log_4 x \):\[ \log_4 9 + \log_4 x = \log_4 (9 \times x) \]

Step 4: Write as a Single Logarithm

The expression is simplified to \( \log_4 (9x) \). This represents the original sum as a single logarithm.

Key Concepts

Product Property of Logarithms

The Product Property of Logarithms is essential when dealing with sums involving logarithms sharing the same base. This property allows us to simplify expressions by transforming the sum of logarithms into a single logarithm. Here's how it works: if you have two logarithmic terms like \( \log_b a \) and \( \log_b c \), they can be added together to create \( \log_b (a \cdot c) \). This is only possible when the logarithms have the same base.

Understanding this property helps in simplifying potentially complex equations and expressions, making it easier to work with them in practical scenarios. Remember, the key is recognizing that both terms share the same "log base", a crucial step before applying the product property.

Logarithmic Expressions

Logarithmic expressions often appear in problems requiring simplification or transformation. These expressions involve the logarithm function, which is the inverse of exponentiation. In the context of our exercise, \( \log_4 9 \) and \( \log_4 x \) are logarithmic expressions.

When dealing with logarithmic expressions, it's important to remember that logarithms have properties like the product, quotient, and power rules. These properties allow you to manipulate logarithmic expressions to make them simpler or to combine multiple logs into a single term.

In solving problems with logarithmic expressions, identifying similarities like the base of the logarithms plays a crucial role and sets the stage for applying rules like the product property.

Single Logarithm

Transforming a sum of logarithms into a single logarithm simplifies the problem. Instead of handling two separate terms, you now have one concise expression. In our example, combining \( \log_4 9 \) and \( \log_4 x \) into \( \log_4 (9x) \) uses the product property effectively.

This transformation is beneficial not only for simplicity but also for computational efficiency, especially in larger and more complex problems. A single logarithmic expression is generally easier to evaluate, derive, or integrate, depending on the objective of your analysis.

By achieving a single logarithm, you often arrive at a more elegant solution, aligning with mathematical simplicity and clarity, which are always preferred in problem-solving. This concept underscores the elegance and power of understanding logarithmic properties.