Solve: \(2^{x-7} \times 5^{x-4} = 1250\)
Solution
\[ 2^{x-7} \times 5^{x-4} = 1250 \]
\[ \Rightarrow 2^{x-7} \times 5^{x-4} = 2 \times 5^4 \]
\[ \Rightarrow \frac{2^x}{2^7} \times \frac{5^x}{5^4} = 2 \times 5^4 \]
\[ \Rightarrow \frac{2^x \cdot 5^x}{2^7 \cdot 5^4} = 2 \cdot 5^4 \]
\[ \Rightarrow 2^x \cdot 5^x = 2^8 \cdot 5^8 \]
\[ \Rightarrow (2 \cdot 5)^x = (2 \cdot 5)^8 \]
\[ \Rightarrow 10^x = 10^8 \]
\[ \Rightarrow x = 8 \]
Final Answer:
\[ \boxed{x = 8} \]