- Logarithmic rules (product, quotient, power) are derived from basic laws of exponents.
- The product rule \(\log_b(MN) = \log_b(M) + \log_b(N)\) utilizes the exponent property \(b^x \cdot b^y = b^{x+y}\).
- Quotient and power rules rely on extensions of this additive property for exponents (e.g., negative exponents and repeated multiplication).
- The correct property from the list used in these proofs is \(\boxed{b^x \cdot b^y = b^{x+y}}\).
Explanation
- Analyzing Logarithmic Rule Proofs
To determine which property is used in the proofs of the product, quotient, and power rules of logarithms, we first recall the definition of a logarithm: \(\log_b(x) = y\) is equivalent to \(b^y = x\). The proofs of the three fundamental logarithmic rules are derived from the laws of exponents.
- Product Rule: \(\log_b(MN) = \log_b(M) + \log_b(N)\). To prove this, let \(M = b^x\) and \(N = b^y\). Then \(MN = b^x \cdot b^y = b^{x+y}\). Applying the definition of a logarithm, \(\log_b(MN) = x + y = \log_b(M) + \log_b(N)\). This proof fundamentally relies on the law of exponents: \(b^x \cdot b^y = b^{x+y}\).
- Quotient Rule: \(\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)\). Letting \(M = b^x\) and \(N = b^y\), we have \(\frac{M}{N} = \frac{b^x}{b^y} = b^x \cdot b^{-y} = b^{x-y}\). By the definition of a logarithm, \(\log_b(\frac{M}{N}) = x - y = \log_b(M) - \log_b(N)\). This also relies on the law of adding exponents: \(b^x \cdot b^{-y} = b^{x-y}\).
- Power Rule: \(\log_b(M^n) = n \cdot \log_b(M)\). Letting \(M = b^x\), then \(M^n = (b^x)^n = b^{nx}\). By the definition, \(\log_b(M^n) = nx = n \cdot \log_b(M)\). This relies on the power of a power exponent rule \((b^x)^n = b^{nx}\), which itself is conceptually rooted in repeated multiplication (e.g., \(b^x \cdot b^x \cdot ... \cdot b^x\) (\(n\) times) equals \(b^{x+x+...+x} = b^{nx}\)).
Therefore, the exponent addition property \(b^x \cdot b^y = b^{x+y}\) is the foundational property used to derive all these rules.
- Evaluating Given Options
The options provided are:
- \(b^x \cdot b^y = b^{x+y}\)
- \(\log_3(M^3) = y \cdot \log_3(M)\)
- \(b^{\frac{x}{y}} = b^{x-y}\)
- \(\log_3(b^y) = y\)
Comparing our derived exponent properties with the given options, the property \(b^x \cdot b^y = b^{x+y}\) is the general law of exponents that justifies addition and subtraction of logarithms (via negative exponents) and, by extension, repeated addition for power laws. The other options are either specific cases, incorrect representations (like the third option containing a typo), or definitions of logarithms. Thus, the correct property is \(b^x \cdot b^y = b^{x+y}\).
Examples
Logarithmic and exponential properties are essential in fields like finance and science for modeling growth and decay. For instance, when calculating compound interest \(A = P(1+r)^t\), taking the logarithm transforms the exponential growth into a linear relationship, allowing researchers to easily isolate time or interest rate variables using the power rule. Understanding these fundamental proofs ensures that complex models can be accurately manipulated and solved in real-world scenarios.