Solve for \( x \) \( 10^{5 x-3}=9^{9 x-5} \) \( x=\square \) You may enter the exact value or round to 4 decimal places. \( > \) Next Question

To solve the equation \( 10^{5x-3} = 9^{9x-5} \), we can take logarithms on both sides. Using the natural logarithm, we rewrite the equation as: \[ (5x - 3) \ln(10) = (9x - 5) \ln(9) \] Expanding both sides: \[ 5x \ln(10) - 3 \ln(10) = 9x \ln(9) - 5 \ln(9) \] Rearranging gives: \[ 5x \ln(10) - 9x \ln(9) = -5 \ln(9) + 3 \ln(10) \] Factoring out \( x \) on the left side: \[ x (5 \ln(10) - 9 \ln(9)) = 3 \ln(10) - 5 \ln(9) \] To isolate \( x \), we divide both sides by \( (5 \ln(10) - 9 \ln(9)) \): \[ x = \frac{3 \ln(10) - 5 \ln(9)}{5 \ln(10) - 9 \ln(9)} \] Now, we can calculate the values of \( \ln(10) \) and \( \ln(9) \): - \( \ln(10) \approx 2.3026 \) - \( \ln(9) \approx 2.1972 \) Substituting in these values: Calculate \( x \): \[ x \approx \frac{3(2.3026) - 5(2.1972)}{5(2.3026) - 9(2.1972)} \\ x \approx \frac{6.9078 - 10.986}{11.513 - 19.7748} \\ x \approx \frac{-4.0782}{-8.2618} \approx 0.494 \] Thus, the approximate solution to four decimal places is: \[ x \approx 0.4940 \]