Problem 4 - Algebra A Combined Function

Question

Write each sum as a single logarithm. Assume that variables represent positive numbers. $$ \log _{2} x+\log _{2} y $$

Step-by-Step Solution

Answer

The sum \( \log_{2} x + \log_{2} y \) is \( \log_{2} (xy) \).

Step 1: Identify the logarithmic properties

We start solving the problem by identifying which properties of logarithms can be applied. One important property is the product rule for logarithms: \( \log_b M + \log_b N = \log_b (MN) \). This property states that the sum of two logarithms with the same base can be combined into a single logarithm by multiplying the arguments.

Step 2: Apply the product rule

Given the expression \( \log_{2} x + \log_{2} y \), we apply the product rule of logarithms. By the rule, we can combine the two logarithms into one: \[ \log_{2} x + \log_{2} y = \log_{2} (xy) \].

Step 3: Write the final expression

Now that we have applied the product rule, the sum of the logarithms \( \log_{2} x + \log_{2} y \) is written as a single logarithm: \( \log_{2} (xy) \). No further simplification is possible, as we have already combined the two logarithms into one.

Key Concepts

Product Rule for Logarithms

The product rule for logarithms is a helpful tool in simplifying expressions that involve logarithms. It states that the sum of two logarithms with the same base can be rewritten as a single logarithm whose argument is the product of the two individual arguments. This is represented by the formula: \( \log_b M + \log_b N = \log_b (MN) \). This means whenever you see two logs of the same base being added, you can "combine" them into a single log by multiplying what's inside. This simplifies calculations and helps make expressions more concise, especially when dealing with larger, complex problems.

Logarithmic Properties

Logarithmic properties are rules that help in manipulating and simplifying logarithms. These properties are fundamental and apply universally across all logarithmic expressions:

• Product Rule: When adding logs with the same base, you can multiply what's inside the logs: \( \log_b M + \log_b N = \log_b (MN) \).
• Quotient Rule: When subtracting logs with the same base, you can divide what's inside: \( \log_b M - \log_b N = \log_b \left(\frac{M}{N}\right) \).
• Power Rule: When you have a power inside a log, you can bring it out front: \( \log_b (M^n) = n \log_b M \).

Understanding these properties allows you to simplify expressions and solve equations involving logarithms effectively. All these rules hinge on basic algebraic principles and serve as the building blocks for more advanced logarithmic operations.

Combined Logarithms

Combined logarithms are expressions where multiple log terms are fused into a singular, simplified logarithmic expression. Using logarithmic properties like the product, quotient, and power rules helps in this process. For instance, the exercise using \( \log_2 x + \log_2 y \) is simplified by combining the two logs into one: \( \log_2 (xy) \). This process is crucial for making equations easier and more straightforward to solve. By combining logarithms, you reduce complexity, which is especially useful in calculus, solving exponential equations, or even when simplifying log-based calculations in everyday problems.