Problem 76 - Algebra A Combined Function

Question

Determine whether each statement is true or false. $$ \log _{7} \frac{14}{8}=\log _{7} 14-\log _{7} 8 $$

Step-by-Step Solution

Answer

True, the equation holds by the logarithm quotient property.

Step 1: Understand the Logarithm Property

The property of logarithms that might be relevant here is the quotient property. It states that for logarithms, \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). We need to check whether this property holds for the given equation.

Step 2: Apply Logarithm Quotient Rule

Using the quotient property of logarithms, we apply it to the left-hand side of the equation, \( \log_7 \left( \frac{14}{8} \right) \). According to the quotient rule, \( \log_7 \left( \frac{14}{8} \right) = \log_7 14 - \log_7 8 \), which matches the right-hand side of the given equation.

Step 3: Evaluate Both Sides

Since we have applied the property correctly, both sides of the equation \( \log_7 \left( \frac{14}{8} \right) = \log_7 14 - \log_7 8 \) are equivalent. Thus, they are the same in value, confirming the property applies correctly.

Step 4: Conclusion

Both sides of the equation are equal when applying the logarithmic property accurately. Therefore, the original statement given in the problem is true based on the application of the logarithmic quotient rule.

Key Concepts

Quotient Property of Logarithms

The quotient property of logarithms is an important fundamental concept when dealing with logarithmic expressions. It allows us to simplify complex logarithmic terms, especially when they involve division. This property states that for any positive numbers \( M \) and \( N \), and a base \( b \), the logarithm of their quotient is equal to the difference of their logarithms: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).

In the context of our original exercise, when given \( \log_7 \left( \frac{14}{8} \right) \), applying the quotient property helps break down the expression into easier parts, \( \log_7 14 \) and \( \log_7 8 \). This makes it possible to determine if the expression is equivalent to \( \log_7 14 - \log_7 8 \), which happens to be true in this case.

This property is particularly useful in simplifying expressions in mathematics and solving logarithmic equations, as it can turn a division problem into a subtraction problem, often making calculations easier.

Logarithm Rules

Logarithms follow specific rules that make solving problems easier. The most common are:

• Product Rule: \( \log_b (MN) = \log_b M + \log_b N \)
• Quotient Rule: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \)
• Power Rule: \( \log_b (M^p) = p \cdot \log_b M \)

Understanding these rules is crucial for correctly solving logarithmic equations. They allow you to manipulate the position of terms, which is often necessary for isolating variables or simplifying expressions.

In our specific example from the exercise, the quotient rule was applied. This rule effectively transforms a division within a logarithm into a simpler subtraction problem. By applying these rules, complex logarithmic expressions become much more manageable, paving the way for accurate solutions.

Mathematical Proof

When solving mathematical problems, especially those involving logarithms, a mathematical proof can provide the verification of certain properties or rules. A mathematical proof involves demonstrating that a certain property or solution holds true universally, under the conditions defined.

In our exercise, proving the statement \( \log_7 \left( \frac{14}{8} \right) = \log_7 14 - \log_7 8 \) involved utilizing the quotient property of logarithms. By breaking down the left side of the equation using the quotient rule, we directly showed it equals the right side, proving the expression true.

Such proofs may seem simple for straightforward rules like the logarithm ones, but they build the foundation of correct logic and reasoning. This process reinforces the significance of properties like the quotient rule and ensures their consistent application across different problems. Doing proofs also enhances understanding and allows one to see not just the 'how' but the 'why' behind mathematical properties.