Problem 74 - Algebra A Combined Function

Question

Determine whether each statement is true or false. $$ \log _{3}(x+y)=\log _{3} x+\log _{3} y $$

Step-by-Step Solution

Answer

False, because the logarithmic property applies to multiplication, not addition.

Step 1: Understand the Given Problem

The problem asks us to determine the validity of the equation \( \log_{3}(x+y) = \log_{3} x + \log_{3} y \). We need to recall the properties of logarithms to verify if the equation holds true.

Step 2: Recall Relevant Logarithmic Property

One important logarithmic property is the product rule: \( \log_{b}(mn) = \log_{b} m + \log_{b} n \). However, for this equation to apply, it requires multiplication inside the logarithm, not addition as in the provided statement.

Step 3: Apply the Product Rule Appropriately

Since the statement involves addition inside the logarithm (\( x+y \)), not multiplication (\( x \times y \)), the product rule of logarithms does not apply here. For the given case \( \log_{3}(x+y) e \log_{3} x + \log_{3} y \). This is because logarithmically combining or splitting terms is reserved for multiplicative expressions.

Step 4: Evaluate the Statement's Validity

Given that the conditions for applying the product rule do not match, the original statement \( \log_{3}(x+y) = \log_{3} x + \log_{3} y \) is incorrect. Thus, the statement is false.

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a key concept in understanding how logarithms work when multiplying numbers. This rule states that for any positive numbers \(m\) and \(n\), and a logarithmic base \(b\), the logarithm of a product is equal to the sum of the logarithms of its factors: \[ \log_{b}(mn) = \log_{b} m + \log_{b} n \] This means that if you are multiplying two numbers inside a logarithm, you can split it into the sum of two separate log expressions. It's important to note that this rule only applies to multiplication inside the logarithm.

• This means that it does not work for addition, subtraction, or any other operations.
• It is a very powerful tool when solving logarithmic equations, as it allows for simplification.

For example, \( \log_{10}(4 \times 5) \) can be rewritten as \( \log_{10} 4 + \log_{10} 5 \). Recognizing these rules is essential in efficiently and accurately managing logarithmic calculations.

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms in representing mathematical quantities. They are part of a system developed to undo exponential functions. At the core, a logarithm asks the question: "To what power must the base be raised to produce a given number?" In a logarithmic expression like \( \log_{b} x \), it indicates the power you must raise \(b\) to in order to get \(x\).

• These expressions can often look complex due to their unique properties and rules.
• Expressions within logarithms can be simplified by applying the laws of logarithms, such as the product, quotient, and power rules.
• Understanding these expressions allows one to solve equations that model real-world scenarios, like calculating compound interest or decay rates.

Thus, appreciating the structure and interpretation of logarithmic expressions is central to harnessing the full potential of logarithmic functions.

Validity of Logarithmic Equations

The validity of logarithmic equations refers to whether the equations accurately reflect the mathematical principles of logarithms. Ensuring validity involves checking if the expression or equation adheres to logarithmic laws and properties. This includes:

• Ensuring proper application of the product, quotient, and power rules.
• Verifying the allowed operations within the logarithmic expressions.

For example, consider the equation \( \log_{3}(x+y) = \log_{3} x + \log_{3} y \). To verify its validity, you would check if the product rule or any other rule applies. In this case, since the product rule requires multiplication and the expression uses addition inside the logarithm, the equation is invalid. It is crucial to review each part of a logarithmic equation to ensure no misapplication occurs. This helps prevent errors in algebraic manipulations and false deductions. Ensuring the validity of equations is not just a theoretical exercise—it has practical implications in accurate computation and in the field's broader applications.