5. Match the exponent equality to the correct solution for \( x \). \( 16^{x+2}=4^{x-2} \) 3. \( 25^{x-5}=5 \) 4. \( 27^{x+3}=5^{x+2} \)

**Problem 5.1: \(16^{x+2}=4^{x-2}\)** 1. Express 16 and 4 as powers of 2: \[ 16 = 2^4 \quad \text{and} \quad 4 = 2^2 \] 2. Rewrite the equation: \[ (2^4)^{x+2} = (2^2)^{x-2} \quad \Longrightarrow \quad 2^{4(x+2)} = 2^{2(x-2)} \] 3. Equate the exponents: \[ 4(x+2) = 2(x-2) \] 4. Solve the linear equation: \[ 4x + 8 = 2x - 4 \quad \Longrightarrow \quad 4x - 2x = -4 - 8 \quad \Longrightarrow \quad 2x = -12 \] \[ x = -6 \] --- **Problem 5.2: \(25^{x-5}=5\)** 1. Express 25 as a power of 5: \[ 25 = 5^2 \] 2. Rewrite the equation: \[ (5^2)^{x-5} = 5 \quad \Longrightarrow \quad 5^{2(x-5)} = 5^1 \] 3. Equate the exponents: \[ 2(x-5) = 1 \] 4. Solve the linear equation: \[ 2x - 10 = 1 \quad \Longrightarrow \quad 2x = 11 \] \[ x = \frac{11}{2} \] --- **Problem 5.3: \(27^{x+3}=5^{x+2}\)** 1. Express 27 as a power of 3: \[ 27 = 3^3 \] 2. Rewrite the equation: \[ (3^3)^{x+3} = 5^{x+2} \quad \Longrightarrow \quad 3^{3(x+3)} = 5^{x+2} \] 3. Take the natural logarithm of both sides: \[ \ln\left(3^{3(x+3)}\right) = \ln\left(5^{x+2}\right) \] 4. Use the power rule of logarithms: \[ 3(x+3)\ln 3 = (x+2)\ln 5 \] 5. Distribute and collect like terms: \[ 3x\ln 3 + 9\ln 3 = x\ln 5 + 2\ln 5 \] \[ 3x\ln 3 - x\ln 5 = 2\ln 5 - 9\ln 3 \] 6. Factor out \( x \): \[ x(3\ln 3 - \ln 5) = 2\ln 5 - 9\ln 3 \] 7. Solve for \( x \): \[ x = \frac{2\ln 5 - 9\ln 3}{3\ln 3 - \ln 5} \] --- **Final Answers:** 1. For \( 16^{x+2}=4^{x-2} \), \( x = -6 \). 2. For \( 25^{x-5}=5 \), \( x = \frac{11}{2} \). 3. For \( 27^{x+3}=5^{x+2} \), \( x = \frac{2\ln 5 - 9\ln 3}{3\ln 3 - \ln 5} \).