To determine which property of logarithms and exponents is used in all the proofs for the product, quotient, and power rules of logarithms, we need to examine each given property and their roles in these proofs.
1. Property 1: [tex]\( b^x \cdot b^y = b^{x+y} \)[/tex]
- This is the product rule for exponents. When multiplying similar bases, we add the exponents.
- This rule is often fundamental in proofs related to logarithms because logarithms are inherently related to exponents.
2. Property 2: [tex]\( \log_3(M^y) = y \cdot \log_3(M) \)[/tex]
- This is the power rule for logarithms, which states that the log of a number raised to a power is that power times the log of the number.
- Essential for simplifying expressions where variables are exponents.
3. Property 3: [tex]\( b^{\frac{x}{y}} = b^{(x-y)} \)[/tex]
- This looks like a misrepresentation. Typically, the quotient rule for exponents is [tex]\( b^{x/y} = (b^x)^{1/y} \)[/tex] or [tex]\( b^{x/y} = (b^x)^{1/y} \neq b^{(x-y)} \)[/tex]. Assuming it's a typo, it doesn't fit.
4. Property 4: [tex]\( \log_3(b^y) = y \)[/tex]
- This doesn't align correctly with known logarithmic properties and is likely another misrepresentation. The appropriate rule is [tex]\( \log_b(b^y) = y \)[/tex].
Given the provided properties and their typical applications:
In the case of verifying proofs for logarithmic rules:
- For the product rule, [tex]\(\log_b (MN) = \log_b M + \log_b N\)[/tex], we rely on the product rule for exponents.
- For the quotient rule, [tex]\(\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\)[/tex], which also involves the product rule for exponents through factoring into a product of a base raised to an exponent.
- For the power rule, [tex]\(\log_b (M^y) = y \cdot \log_b M\)[/tex], the property of exponents is directly used.
The fundamental property used across all these instances is indeed the first property:
[tex]\[ b^x \cdot b^y = b^{x+y} \][/tex]
Conclusion: The property used in all the proofs for logarithm rules is:
[tex]\[ b^x \cdot b^y = b^{x+y} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
