\(4 < a < 5\)
\(3 < a < 4\)
\(a > 5\)
\(2 < a < 3\)
Given:
\[ \frac{\log_{15} a + \log_{32} a}{(\log_{15} a)(\log_{32} a)} = 4 \]
Step 1: Change base to common logarithm
Apply the change of base formula \( \log_{b} a = \frac{\log a}{\log b} \): \[ \frac{\frac{\log a}{\log 15} + \frac{\log a}{\log 32}}{\left(\frac{\log a}{\log 15}\right) \left(\frac{\log a}{\log 32}\right)} \]
Step 2: Simplify the expression
\[ \frac{\log a \left( \frac{1}{\log 15} + \frac{1}{\log 32} \right)}{\frac{(\log a)^2}{\log 15 \cdot \log 32}} \] \[ = \frac{\log a \left( \frac{\log 32 + \log 15}{\log 15 \cdot \log 32} \right)}{\frac{(\log a)^2}{\log 15 \cdot \log 32}} \] \[ = \frac{\log a (\log 32 + \log 15)}{(\log a)^2} \] \[ = \frac{\log 32 + \log 15}{\log a} \]
Step 3: Solve for \( a \) using the original equation
\[ \frac{\log 32 + \log 15}{\log a} = 4 \] This implies \[ \log 32 + \log 15 = 4 \log a \]. Using the logarithm property \( \log m + \log n = \log (mn) \): \[ \log(32 \cdot 15) = 4 \log a \] \[ \log 480 = \log a^4 \Rightarrow a^4 = 480 \]
Step 4: Estimate the value of \( a \)
Consider powers of 4:
\[ 4^4 = 256 \quad \text{and} \quad 5^4 = 625 \] Since \( 256<480<625 \), it follows that \( 4<a<5 \).
Final Answer: Option (A): \( \boxed{4 < a < 5} \)
If \[ \log_{p^{1/2}} y \times \log_{y^{1/2}} p = 16, \] then find the value of the given expression.