To decide which property of logarithms and exponents is used in the proofs of the product, quotient, or power rules of logarithms, we need to evaluate each property carefully.
1. [tex]\( b^x \cdot b^y = b^{x+y} \)[/tex]
- This property is fundamental for the product of exponents when the bases are the same. While it is useful in the derivation of logarithmic rules, it is not the defining property for logarithms.
2. [tex]\( \log _3\left(M M^\nu\right)=y \cdot \log _8(M) \)[/tex]
- This expression appears to be incorrect or misinterpreted. The term on the left-hand side does not logically simplify into the given right-hand side. Therefore, this can't be the universally used property in proofs.
3. [tex]\( b^{\frac{x}{y}}=b^{(x-y)} \)[/tex]
- This property is incorrect as it states a false identity between exponential expressions.
4. [tex]\( \log _b\left(b^y\right)=y \)[/tex]
- This property is a fundamental identity in logarithms. It states that the logarithm of a number to its own base is the exponent itself. This property is crucial and is used in simplifying and validating the results in the proofs of the logarithmic rules.
Given the definitions and the properties usually invoked in logarithmic operations and proofs:
- Product Rule for Logarithms: [tex]\(\log_b(MN) = \log_b(M) + \log_b(N)\)[/tex]
- Quotient Rule for Logarithms: [tex]\(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\)[/tex]
- Power Rule for Logarithms: [tex]\(\log_b(M^k) = k \cdot \log_b(M)\)[/tex]
Each proof involves returning to the fundamental identity that [tex]\(\log_b(b^y) = y\)[/tex].
Thus, the property that is used in all proof scenarios for product, quotient, or power rules of logarithms is:
[tex]\[ \boxed{\log _b\left(b^y\right)=y} \][/tex]
