Problem 126 - College Algebra

Question

In Exercises \(125-128,\) determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \frac{\log _{7} 49}{\log _{7} 7}=\log _{7} 49-\log _{7} 7 $$

Step-by-Step Solution

Answer

False. The correct statement should be \(\frac{\log _{7} 49}{\log _{7} 7} = \log _{7} 49 - \log _{7} 7 + \log _{7} 7\).

Step 1: Understand and Apply Logarithm Properties

Use the properties of logarithms to simplify both sides of the equation. For the left-hand side, recall that \(\frac{\log _{b} a}{\log _{b} c} = \log _{c} a\). So, in our case, \(\frac{\log _{7} 49}{\log _{7} 7} = \log _{7} 49 = 2\), since \(7^2 = 49\). For the right-hand side, we remember the property that states \(\log _{b}(a) - \log _{b} (c) = \log _{b} (\frac{a}{c})\). Using these properties, our expression becomes \(\log _{7} 49 - \log _{7} 7 = \log _{7} (\frac{49}{7}) = \log _{7} 7 = 1\).

Step 2: Compare Both Sides

Compare both sides to evaluate the validity of the given statement. It turns out \(2 ≠ 1\). Hence, the given expression is false.

Step 3: Make necessary changes to produce a true statement

Having determined that the initial statement is false, we need to change things around to render it true. One way of doing this is by adjusting the right-hand side to match the left-hand side. We can do this by adding a 'log 7' term: \(\frac{\log _{7} 49}{\log _{7} 7} = \log _{7} 49 - \log _{7} 7 + \log _{7} 7\). Now after simplification, both sides will equate to 2, rendering the modified statement true.

Key Concepts

Logarithm Properties

Logarithms are mathematical expressions used to represent exponents that a base number must be raised to, to produce a given number. Understanding properties of logarithms helps in breaking down complex logarithmic problems. Here are some key properties:

• Quotient Rule: This rule states that \( \frac{\log_b a}{\log_b c} = \log_c a \). This property allows us to simplify logarithmic expressions by changing the base.
• Difference Rule: Given two logs with the same base, \( \log_b (a) - \log_b (c) = \log_b \left(\frac{a}{c}\right) \). This helps in solving logarithmic equations by turning a subtraction of logarithms into a division of their arguments.

By applying these properties, we can reconfigure complex logarithmic expressions into simpler terms, facilitating better analysis and solutions. Grasping these rules is key to solving logarithmic equations efficiently. It helps to remember that properties of exponents and logarithms are intimately connected. Both seek to simplify and break down mathematical expressions into more manageable components.

Equation Solving

Equation solving with logarithms involves manipulating logarithmic expressions to either simplify them or to make both sides of the equation equal. In our problem, we had to evaluate an equation to discern its truthfulness.

• First, we simplified both sides of the equation using logarithm properties.
• For the left term \( \frac{\log_7 49}{\log_7 7} \), using the quotient rule, we found it equals \( 2 \) because \( 7^2 = 49 \).
• The right term \( \log_7 49 - \log_7 7 \) simplifies to \( \log_7\left(\frac{49}{7}\right) = \log_7 7 = 1 \), using the difference rule.

After simplification, solving the equation calls for comparison, leading us to discover that \( 2 eq 1 \), and thus the statement was false. Solving logarithmic equations often requires identifying and applying appropriate properties to either simplify terms or convert the expression into a form where both sides balance.

Mathematical Statements Evaluation

In this type of evaluation, the goal is to determine the truth of a mathematical statement. If a statement is false, one must adjust it to make it true. When working with logarithmic equations, this involves ensuring identities match through valid algebraic manipulation.

• Identifying Discrepancy: It is important to start by working on both sides of the equation separately to simplify and identify where discrepancies exist.
• Adjusting Equivalence: The goal is to adjust the false equation to one that holds true, often by reconfiguring components so each side equals.
• Addition/Subtraction: Introducing terms to counterbalance discrepancies might involve adding, subtracting, or rearranging logarithms.

For the given problem, the discovery that \(2 eq 1\) led to the realization that the initial statement was incorrect as given. We rectified it by adjusting the right-hand side of the equation with additional terms until balance was restored, making the statement correct. Learning how to evaluate and correct mathematical statements hones analytical skills crucial for higher-level mathematics.