Question: (5) \( 25^{x-1} = 125 \) (6) \( 9^{2x+1} = 81 \)

Solution

First, express both sides with the same base. \[ 25 = 5^2 \quad \text{and} \quad 125 = 5^3 \] Substitute these into the equation: \[ (5^2)^{x-1} = 5^3 \] Simplify the left side: \[ 5^{2(x-1)} = 5^3 \] Since the bases are the same, set the exponents equal: \[ 2(x - 1) = 3 \] Solve for \(x\): \[ 2x - 2 = 3 \] \[ 2x = 5 \] \[ x = \frac{5}{2} \]