Solve for \( x \) given that \( 5^{2 x+2}-20 \times 5^{2 x}=575 \)

To solve the equation \( 5^{2x+2} - 20 \times 5^{2x} = 575 \), we start by simplifying the equation. First, notice that \( 5^{2x+2} = 5^{2x} \cdot 5^2 = 25 \cdot 5^{2x} \). This allows us to rewrite the equation as: \[ 25 \cdot 5^{2x} - 20 \cdot 5^{2x} = 575 \] Now combine like terms: \[ (25 - 20) \cdot 5^{2x} = 575 \] This simplifies to: \[ 5 \cdot 5^{2x} = 575 \] Next, divide both sides by 5: \[ 5^{2x} = \frac{575}{5} = 115 \] Now we take the logarithm base 5 of both sides to solve for \( 2x \): \[ 2x = \log_5(115) \] To isolate \( x \): \[ x = \frac{1}{2} \log_5(115) \] If you want to further approximate \( x \), you could convert this to base 10 or natural logarithm using the change of base formula: \[ x \approx \frac{1}{2} \cdot \frac{\log_{10}(115)}{\log_{10}(5)} \] Calculating this gives: \[ \log_{10}(115) \approx 2.0607 \quad \text{and} \quad \log_{10}(5) \approx 0.6990 \] Thus, \[ x \approx \frac{1}{2} \cdot \frac{2.0607}{0.6990} \approx \frac{1}{2} \cdot 2.949 = 1.4745 \] Therefore, the solution to the equation is: \[ x \approx 1.4745 \]