(3) \( 3^{x+1}-2 \cdot 3^{x-1}=21 \) (5) \( 3^{2 x}-4 \cdot 3^{x}+3=0 \) (7) \( 3^{2 x+1}-4 \cdot 3^{x}+1=0 \) (9) \( 3^{x}-4=\frac{45}{3^{x}} \) (11) \( 2^{x}-3 \cdot 2^{3-x}-5=0 \)

Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x+1}-2\times 3^{x-1}=21\) - step1: Subtract the terms: \(7\times 3^{x-1}=21\) - step2: Divide both sides: \(\frac{7\times 3^{x-1}}{7}=\frac{21}{7}\) - step3: Divide the numbers: \(3^{x-1}=3\) - step4: Set the exponents equal: \(x-1=1\) - step5: Move the constant to the right side: \(x=1+1\) - step6: Add the numbers: \(x=2\) Solve the equation \( 3^{2 x+1}-4 \cdot 3^{x}+1=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{2x+1}-4\times 3^{x}+1=0\) - step1: Factor the expression: \(\left(3^{x}-1\right)\left(3^{x+1}-1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&3^{x}-1=0\\&3^{x+1}-1=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=0\\&x=-1\end{align}\) - step4: Rewrite: \(x_{1}=-1,x_{2}=0\) Solve the equation \( 2^{x}-3 \cdot 2^{3-x}-5=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{x}-3\times 2^{3-x}-5=0\) - step1: Factor the expression: \(\left(2^{x}-8\right)\left(2^{x}+3\right)\left(2^{x}\right)^{-1}=0\) - step2: Rewrite the expression: \(\frac{2^{2x}-5\times 2^{x}-24}{2^{x}}=0\) - step3: Cross multiply: \(2^{2x}-5\times 2^{x}-24=2^{x}\times 0\) - step4: Simplify the equation: \(2^{2x}-5\times 2^{x}-24=0\) - step5: Factor the expression: \(\left(2^{x}-8\right)\left(2^{x}+3\right)=0\) - step6: Separate into possible cases: \(\begin{align}&2^{x}-8=0\\&2^{x}+3=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=3\\&x \notin \mathbb{R}\end{align}\) - step8: Find the union: \(x=3\) Solve the equation \( 3^{2 x}-4 \cdot 3^{x}+3=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{2x}-4\times 3^{x}+3=0\) - step1: Factor the expression: \(\left(3^{x}-3\right)\left(3^{x}-1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&3^{x}-3=0\\&3^{x}-1=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=1\\&x=0\end{align}\) - step4: Rewrite: \(x_{1}=0,x_{2}=1\) Solve the equation \( 3^{x}-4=\frac{45}{3^{x}} \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x}-4=\frac{45}{3^{x}}\) - step1: Divide the terms: \(3^{x}-4=5\times 3^{2-x}\) - step2: Move the expression to the left side: \(3^{x}-4-5\times 3^{2-x}=0\) - step3: Factor the expression: \(\left(3^{x}-9\right)\left(3^{x}+5\right)\left(3^{x}\right)^{-1}=0\) - step4: Rewrite the expression: \(\frac{3^{2x}-4\times 3^{x}-45}{3^{x}}=0\) - step5: Cross multiply: \(3^{2x}-4\times 3^{x}-45=3^{x}\times 0\) - step6: Simplify the equation: \(3^{2x}-4\times 3^{x}-45=0\) - step7: Factor the expression: \(\left(3^{x}-9\right)\left(3^{x}+5\right)=0\) - step8: Separate into possible cases: \(\begin{align}&3^{x}-9=0\\&3^{x}+5=0\end{align}\) - step9: Solve the equation: \(\begin{align}&x=2\\&x \notin \mathbb{R}\end{align}\) - step10: Find the union: \(x=2\) Here are the solutions for each of the equations provided: 1. **Equation (3)**: \( 3^{x+1}-2 \cdot 3^{x-1}=21 \) - **Solution**: \( x = 2 \) 2. **Equation (5)**: \( 3^{2 x}-4 \cdot 3^{x}+3=0 \) - **Solutions**: \( x_{1} = 0 \), \( x_{2} = 1 \) 3. **Equation (7)**: \( 3^{2 x+1}-4 \cdot 3^{x}+1=0 \) - **Solutions**: \( x_{1} = -1 \), \( x_{2} = 0 \) 4. **Equation (9)**: \( 3^{x}-4=\frac{45}{3^{x}} \) - **Solution**: \( x = 2 \) 5. **Equation (11)**: \( 2^{x}-3 \cdot 2^{3-x}-5=0 \) - **Solution**: \( x = 3 \) If you need further explanations or steps for any of these solutions, feel free to ask!