To determine which property or properties of logarithms are used to rewrite the given expression [tex]\(\log _2 14 - \log _2 7 = \log _2 2\)[/tex], we need to analyze the logarithmic statements step by step.
Here is the given logarithmic equation:
[tex]\[
\log _2 14 - \log _2 7 = \log _2 2
\][/tex]
1. Identify the properties of logarithms that might be applicable:
- Quotient Property: [tex]\(\log_b(m) - \log_b(n) = \log_b\left(\frac{m}{n}\right)\)[/tex]
- Product Property: [tex]\(\log_b(mn) = \log_b(m) + \log_b(n)\)[/tex]
- Power Property: [tex]\(\log_b(m^n) = n \log_b(m)\)[/tex]
- (Non-existent) Difference Property: Not an actual logarithmic property
2. Rewrite the expression using the Quotient Property:
According to the Quotient Property of logarithms, which states that the difference of two logarithms with the same base is equivalent to the logarithm of the quotient of their arguments:
[tex]\[
\log _2 14 - \log _2 7 = \log _2 \left(\frac{14}{7}\right)
\][/tex]
3. Simplify the quotient inside the logarithm:
Calculate the argument inside the logarithm:
[tex]\[
\frac{14}{7} = 2
\][/tex]
Thus, the equation simplifies to:
[tex]\[
\log _2 \left(\frac{14}{7}\right) = \log _2 2
\][/tex]
This confirms that the right-hand side and the left-hand side of the equation are indeed equal:
[tex]\[
\log _2 2 = \log _2 2
\][/tex]
4. Conclusion:
The property of logarithms used to transform the expression [tex]\(\log _2 14 - \log _2 7 = \log _2 2\)[/tex] is the Quotient Property.
Therefore, the answer is:
[tex]\[
\boxed{\text{Quotient Property}}
\][/tex]
