Solve for \( x \) : (a) \( 10^{x} \cdot 9^{x}=1000 \) (b) \( 10^{x} \cdot 9^{2 x}=1000 \)

To solve the equations, we will use logarithmic properties. (a) \( 10^{x} \cdot 9^{x} = 1000 \) First, we can rewrite \( 1000 \) as \( 10^3 \): \[ 10^{x} \cdot 9^{x} = 10^3 \] Now, we can express this as: \[ (10 \cdot 9)^{x} = 10^3 \] This simplifies to: \[ 90^{x} = 10^3 \] Taking the logarithm of both sides: \[ x \cdot \log(90) = 3 \cdot \log(10) \] Thus, solving for \( x \): \[ x = \frac{3 \cdot \log(10)}{\log(90)} \approx \frac{3}{1.954} \approx 1.53 \] (b) \( 10^{x} \cdot 9^{2x} = 1000 \) Again, we can rewrite \( 1000 \) as \( 10^3 \): \[ 10^{x} \cdot 9^{2x} = 10^3 \] Express this as: \[ 10^{x} \cdot (9^2)^{x} = 10^3 \] Which means: \[ 10^{x} \cdot 81^{x} = 10^3 \] Now we can combine the bases: \[ (10 \cdot 81)^{x} = 10^3 \] This simplifies to: \[ 810^{x} = 10^3 \] Taking the logarithm of both sides: \[ x \cdot \log(810) = 3 \cdot \log(10) \] Solving for \( x \): \[ x = \frac{3 \cdot \log(10)}{\log(810)} \approx \frac{3}{2.908} \approx 1.03 \] In conclusion, the solutions are approximately: (a) \( x \approx 1.53 \) (b) \( x \approx 1.03 \)