a. \( 4^{2 x-5}=4^{4 x+7} \) \( \begin{array}{ll}\text { c. } 5^{3 x+6}=5^{x-1}=2^{4 x-6} \\ \text { b. } & \text { d. } 81^{x}=3^{5 x+8}\end{array} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(81^{x}=3^{5x+8}\) - step1: Rewrite the expression: \(3^{4x}=3^{5x+8}\) - step2: Set the exponents equal: \(4x=5x+8\) - step3: Move the variable to the left side: \(4x-5x=8\) - step4: Subtract the terms: \(-x=8\) - step5: Change the signs: \(x=-8\) Solve the equation \( 5^{3 x+6}=5^{x-1} \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{3x+6}=5^{x-1}\) - step1: Set the exponents equal: \(3x+6=x-1\) - step2: Move the expression to the left side: \(3x-x=-1-6\) - step3: Add and subtract: \(2x=-1-6\) - step4: Add and subtract: \(2x=-7\) - step5: Divide both sides: \(\frac{2x}{2}=\frac{-7}{2}\) - step6: Divide the numbers: \(x=-\frac{7}{2}\) Solve the equation \( 4^{2 x-5}=4^{4 x+7} \). Solve the equation by following steps: - step0: Solve for \(x\): \(4^{2x-5}=4^{4x+7}\) - step1: Set the exponents equal: \(2x-5=4x+7\) - step2: Move the expression to the left side: \(2x-4x=7+5\) - step3: Add and subtract: \(-2x=7+5\) - step4: Add and subtract: \(-2x=12\) - step5: Change the signs: \(2x=-12\) - step6: Divide both sides: \(\frac{2x}{2}=\frac{-12}{2}\) - step7: Divide the numbers: \(x=-6\) Let's solve each equation step by step. ### a. \( 4^{2x-5} = 4^{4x+7} \) Since the bases are the same, we can set the exponents equal to each other: \[ 2x - 5 = 4x + 7 \] Now, let's solve for \( x \): 1. Rearranging the equation: \[ 2x - 4x = 7 + 5 \] \[ -2x = 12 \] 2. Dividing both sides by -2: \[ x = -6 \] ### b. \( 5^{3x+6} = 5^{x-1} \) Again, since the bases are the same, we can set the exponents equal to each other: \[ 3x + 6 = x - 1 \] Now, let's solve for \( x \): 1. Rearranging the equation: \[ 3x - x = -1 - 6 \] \[ 2x = -7 \] 2. Dividing both sides by 2: \[ x = -\frac{7}{2} \quad \text{or} \quad x = -3.5 \] ### c. \( 81^{x} = 3^{5x+8} \) We can express \( 81 \) as \( 3^4 \): \[ (3^4)^{x} = 3^{5x+8} \] This simplifies to: \[ 3^{4x} = 3^{5x+8} \] Setting the exponents equal to each other: \[ 4x = 5x + 8 \] Now, let's solve for \( x \): 1. Rearranging the equation: \[ 4x - 5x = 8 \] \[ -x = 8 \] 2. Dividing both sides by -1: \[ x = -8 \] ### Summary of Solutions: - a. \( x = -6 \) - b. \( x = -\frac{7}{2} \) or \( x = -3.5 \) - c. \( x = -8 \)