5•2^х+2•5^х=7•10^(х/2)

To solve the equation \(5 \cdot 2^x + 2 \cdot 5^x = 7 \cdot 10^{x/2}\), we can start by rewriting the powers to a common base if possible, or analyzing the equation structurally. Recall that: \[ 10^{x/2} = (2 \cdot 5)^{x/2} = 2^{x/2} \cdot 5^{x/2}. \] Substituting this into the equation gives: \[ 5 \cdot 2^x + 2 \cdot 5^x = 7 \cdot (2^{x/2} \cdot 5^{x/2}). \] Let's simplify the equation step by step. Divide both sides by \(5^{x/2}\) to isolate \(2^{x/2}\): \[ \frac{5 \cdot 2^x}{5^{x/2}} + 2 = 7 \cdot \left(\frac{2^{x/2}}{5^{x/2}}\right), \] which simplifies to: \[ 5 \cdot 2^{x - x/2} + 2 = 7 \cdot \left(\frac{2^{x/2}}{5^{x/2}}\right). \] We can also take a different approach by finding specific values for \(x\) to check for solutions: 1. If \(x = 0\): \[ 5 \cdot 2^0 + 2 \cdot 5^0 = 5 + 2 = 7, \] and \[ 7 \cdot 10^{0/2} = 7 \cdot 1 = 7. \] Therefore, \(x = 0\) is a solution. Next, let’s analyze if there are any other solutions. If we continue simplifying or conducting numeric checks or graphing, we would look at the behavior of both sides of the equation in general. To check for additional roots or intersections, you can graph the equation: \[ y_1 = 5 \cdot 2^x + 2 \cdot 5^x \quad \text{and} \quad y_2 = 7 \cdot 10^{x/2} \] and see where they intersect. Alternatively, conducting tests with other integer values of \(x\) may yield further insight, but it appears that \(x = 0\) may indeed be the only real solution based on the behavior of the exponential growth of respective terms. If you require a deeper investigation or numerical solutions, numerical methods or computational tools can help further analyze the function. For the purpose of this equation, the evident solution is: \[ \boxed{0}. \]