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.
$\frac{lo{g}_{7}49}{lo{g}_{7}7}=lo{g}_{7}49-lo{g}_{7}7$

Verified step by step guidance

1. Recall the logarithm property for division: \( \frac{\log_b A}{\log_b B} \) is not equal to \( \log_b A - \log_b B \). Instead, the subtraction property applies to logarithms of products or quotients inside the log, such as \( \log_b \frac{A}{B} = \log_b A - \log_b B \).
2. Evaluate the left side expression \( \frac{\log_7 49}{\log_7 7} \). Since \( \log_7 7 = 1 \), this simplifies to \( \log_7 49 \).
3. Evaluate the right side expression \( \log_7 49 - \log_7 7 \). Using the subtraction property of logarithms, this equals \( \log_7 \frac{49}{7} = \log_7 7 \).
4. Compare the simplified left side \( \log_7 49 \) and right side \( \log_7 7 \). Since \( 49 \neq 7 \), the two sides are not equal, so the original statement is false.
5. To make the statement true, replace the division on the left side with subtraction inside the logarithm: \( \frac{\log_7 49}{\log_7 7} \) should be replaced by \( \log_7 \frac{49}{7} \), so the true statement is \( \log_7 \frac{49}{7} = \log_7 49 - \log_7 7 \).

Key Concepts

Properties of Logarithms

Logarithms have specific properties that simplify expressions, such as the product, quotient, and power rules. Understanding these properties helps in manipulating and evaluating logarithmic expressions correctly.

Quotient Rule for Logarithms

The quotient rule states that the logarithm of a quotient is the difference of the logarithms: log_b(A/B) = log_b(A) - log_b(B). This rule is essential for rewriting and simplifying expressions involving division inside a logarithm.

Evaluating Logarithms with Known Bases and Arguments

Evaluating logarithms like log7 49 and log7 7 involves recognizing powers of the base: 49 = 7^2 and 7 = 7^1. This allows simplification to numerical values, which is crucial for verifying the truth of logarithmic statements.