Step 1: Understanding the Concept:
When we have terms of the form \( 1/\log_a b \), we can use the base-change property to make the logarithms easier to combine. : Key Formula or Approach:
1. \( \frac{1}{\log_a b} = \log_b a \).
2. \( \log a + \log b + \log c + \log d = \log(abcd) \). Step 2: Detailed Explanation:
Apply the property \( \frac{1}{\log_a b} = \log_b a \) to each term:
\[ S = \log_{10} 25 + \log_{10} 4 + \log_{10} \sqrt{2} + \log_{10} \sqrt{5} \]
Using the product property of logarithms:
\[ S = \log_{10} (25 \times 4 \times \sqrt{2} \times \sqrt{5}) \]
\[ S = \log_{10} (100 \times \sqrt{2 \times 5}) \]
\[ S = \log_{10} (100 \times \sqrt{10}) \]
Express 100 and \( \sqrt{10} \) as powers of 10:
\[ S = \log_{10} (10^2 \times 10^{1/2}) \]
\[ S = \log_{10} (10^{2 + 0.5}) \]
\[ S = \log_{10} (10^{2.5}) = 2.5 = \frac{5}{2} \]. Step 3: Final Answer:
The sum is 5/2.