- Simplify the interior: \(\log_3 8 + \log_3 z = \log_3(8z)\).
- Apply the power rule: \(2 \log_3(8z) = \log_3(64z^2)\).
- Calculate the denominator: \(3^4 - 7^2 = 81 - 49 = 32\).
- Apply the quotient rule: \(\log_3(64z^2) - \log_3(32) = \log_3\left(\frac{64z^2}{32}\right) = \boxed{\log_3 2 z^2}\).
Explanation
- Analysis of the logarithmic expression
We are tasked to simplify the expression \(2(\log_3 8 + \log_3 z) - \log_3(3^4 - 7^2)\) and write it as a single logarithm with base 3. We will utilize the fundamental properties of logarithms:
- Product rule: \(\log_b M + \log_b N = \log_b(MN)\).
- Power rule: \(k \log_b M = \log_b(M^k)\).
- Quotient rule: \(\log_b M - \log_b N = \log_b\left(\frac{M}{N}\right)\).
Additionally, we need to evaluate the constant term \(3^4 - 7^2\).
- Simplifying the first part of the expression
First, we simplify the expression inside the parenthesis. Using the product rule, \(\log_3 8 + \log_3 z = \log_3(8z)\). The expression then becomes \(2 \log_3(8z) - \log_3(3^4 - 7^2)\). Next, applying the power rule, we have \(2 \log_3(8z) = \log_3((8z)^2) = \log_3(64z^2)\).
- Evaluating the constant term
Now, we evaluate the constant term in the second logarithm: \(3^4 = 81\) and \(7^2 = 49\). Subtracting these gives \(3^4 - 7^2 = 81 - 49 = 32\). The full expression is now reduced to \(\log_3(64z^2) - \log_3(32)\).
- Combining and final simplification
Finally, applying the quotient rule \(\log_3 A - \log_3 B = \log_3\left(\frac{A}{B}\right)\), we get:
\[\log_3(64z^2) - \log_3(32) = \log_3\left(\frac{64z^2}{32}\right).\]
Simplifying the fraction \(\frac{64}{32} = 2\), the expression becomes \(\log_3(2z^2)\). This matches one of the provided options. Therefore, the simplified single logarithm is \(\boxed{\log_3 2 z^2}\).
Examples
Logarithmic properties are essential in fields like acoustics to measure sound intensity (decibels) or in chemistry to calculate pH levels, where large ranges of values are compressed into smaller, manageable scales. For example, if an engineer is scaling audio signals in a production environment, they might use the property \(\log_b(M) - \log_b(N) = \log_b(M/N)\) to compare the gain ratios of two separate amplifiers. This ensures that complicated multiplicative relationships between power levels can be handled easily through addition and subtraction, keeping calculations efficient and accurate.