Solving Logarithmic Expressions with Log Rules
To solve the expression given in the image, \( \log_{c^2}(\frac{a^5b}{c^2}) \), we can apply log rules (quotient rule, power rule, and change of base formula).
The quotient rule states that \( \log(\frac{x}{y}) = \log(x) - \log(y) \).
The power rule states that \( \log(x^a) = a \log(x) \).
The change of base formula states that \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \).
Let's break down the expression by using these rules:
1. Apply the quotient rule to the argument of the log.
\( \log_{c^2}(a^5b) - \log_{c^2}(c^2) \)
2. Apply the power rule to \( a^5 \) and \( c^2 \) within the log expressions.
\( 5 \log_{c^2}(a) + \log_{c^2}(b) - 2 \log_{c^2}(c) \)
3. Notice that \( \log_{c^2}(c^2) = 1 \) because the log base and the argument are the same value raised to the same power.
Hence, the expression becomes:
\( 5 \log_{c^2}(a) + \log_{c^2}(b) - 2 \)
4. Apply the change of base formula to the two remaining log terms.
Since we want to express everything in terms of the base c, we get:
\( 5 \frac{\log_{c}(a)}{\log_{c}(c^2)} + \frac{\log_{c}(b)}{\log_{c}(c^2)} - 2 \)
5. Simplify by recognizing that \( \log_{c}(c^2) = 2 \).
\( 5 \frac{\log_{c}(a)}{2} + \frac{\log_{c}(b)}{2} - 2 \)
6. Simplify the fractions.
\( \frac{5}{2} \log_{c}(a) + \frac{1}{2} \log_{c}(b) - 2 \)
7. Looking at the answer choices given, we can see that our derived expression matches choice D when we distribute the 2 outside of the log:
\( \frac{5}{2} \log_{c}(a) + \frac{1}{2} \log_{c}(b) - 2 \cdot 1 \)
\( \frac{5}{2} \log_{c}(a) + \frac{1}{2} \log_{c}(b) - 2 \log_{c}(c) \)
Since \( \log_{c}(c) = 1 \), the 2 just becomes \( 2 \log_{c}(c) \), and thus the answer is:
\( \frac{5}{2} \cdot \log_{c}(a) + \frac{1}{2} \cdot \log_{c}(b) - 2 \log_{c}(c) \)
So, the correct answer is D:
\( 5 \log_{c}(a) + \log_{c}(b) - 2 \log_{c}(c) \)
