OEFENING 5 Los op vir \( x \) : \( 1 \quad 18^{x} \cdot \frac{1}{4^{x-1}}=\frac{1}{32} \) \( 2 \quad 3^{x+1}=9^{x} \cdot \frac{1}{27} \) \( 3 \quad 3^{x^{2}+2}=9 \cdot \frac{1}{27^{x}} \) \( 4 \quad 5^{x^{2}-1} \cdot \frac{1}{25}=25^{x} \) \( 5 \quad 2^{2 x} \cdot 4^{x+1}=1 \) \( 6 \quad 12^{x} .12^{x}=144 \) \( 7 \quad 2^{x^{2}} \cdot 2^{x^{2}-2}+1=65 \) \( 8 \quad \frac{4^{x+1} \times\left(\frac{1}{4}\right)^{-x}}{2^{2 x+1}}=\frac{1}{32} \) \( 9 \quad 81.9^{x-1}=\frac{1}{27} \) \( 10 \quad 10^{2 x^{2}} \cdot 10^{x+1}=10000 \)

Solve the equation by following steps: - step0: Solve for \(x\): \(12^{x}\times 12^{x}=144\) - step1: Multiply the terms: \(12^{2x}=144\) - step2: Rewrite in exponential form: \(12^{2x}=12^{2}\) - step3: Set the exponents equal: \(2x=2\) - step4: Divide both sides: \(\frac{2x}{2}=\frac{2}{2}\) - step5: Divide the numbers: \(x=1\) Solve the equation \( 3^{x+1}=9^{x} \cdot \frac{1}{27} \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x+1}=9^{x}\times \frac{1}{27}\) - step1: Multiply the terms: \(3^{x+1}=3^{2x-3}\) - step2: Set the exponents equal: \(x+1=2x-3\) - step3: Move the expression to the left side: \(x-2x=-3-1\) - step4: Add and subtract: \(-x=-3-1\) - step5: Add and subtract: \(-x=-4\) - step6: Change the signs: \(x=4\) Solve the equation \( 2^{2 x} \cdot 4^{x+1}=1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{2x}\times 4^{x+1}=1\) - step1: Multiply the terms: \(2^{4x+2}=1\) - step2: Rewrite in exponential form: \(2^{4x+2}=2^{0}\) - step3: Set the exponents equal: \(4x+2=0\) - step4: Move the constant to the right side: \(4x=0-2\) - step5: Remove 0: \(4x=-2\) - step6: Divide both sides: \(\frac{4x}{4}=\frac{-2}{4}\) - step7: Divide the numbers: \(x=-\frac{1}{2}\) Solve the equation \( 81 \cdot 9^{x-1}=\frac{1}{27} \). Solve the equation by following steps: - step0: Solve for \(x\): \(81\times 9^{x-1}=\frac{1}{27}\) - step1: Multiply the terms: \(9^{x+1}=\frac{1}{27}\) - step2: Rewrite in exponential form: \(3^{2\left(x+1\right)}=3^{-3}\) - step3: Set the exponents equal: \(2\left(x+1\right)=-3\) - step4: Divide both sides: \(\frac{2\left(x+1\right)}{2}=\frac{-3}{2}\) - step5: Divide the numbers: \(x+1=-\frac{3}{2}\) - step6: Move the constant to the right side: \(x=-\frac{3}{2}-1\) - step7: Subtract the numbers: \(x=-\frac{5}{2}\) Solve the equation \( \frac{4^{x+1} \times \left(\frac{1}{4}\right)^{-x}}{2^{2 x+1}}=\frac{1}{32} \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{4^{x+1}\left(\frac{1}{4}\right)^{-x}}{2^{2x+1}}=\frac{1}{32}\) - step1: Reduce the fraction: \(2^{2x+1}=\frac{1}{32}\) - step2: Rewrite in exponential form: \(2^{2x+1}=2^{-5}\) - step3: Set the exponents equal: \(2x+1=-5\) - step4: Move the constant to the right side: \(2x=-5-1\) - step5: Subtract the numbers: \(2x=-6\) - step6: Divide both sides: \(\frac{2x}{2}=\frac{-6}{2}\) - step7: Divide the numbers: \(x=-3\) Solve the equation \( 3^{x^{2}+2}=9 \cdot \frac{1}{27^{x}} \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x^{2}+2}=9\times \frac{1}{27^{x}}\) - step1: Simplify: \(3^{x^{2}+2}=3^{-3x+2}\) - step2: Set the exponents equal: \(x^{2}+2=-3x+2\) - step3: Cancel equal terms: \(x^{2}=-3x\) - step4: Move the expression to the left side: \(x^{2}-\left(-3x\right)=0\) - step5: Add or subtract both sides: \(x^{2}+3x=0\) - step6: Factor the expression: \(x\left(x+3\right)=0\) - step7: Separate into possible cases: \(\begin{align}&x=0\\&x+3=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=0\\&x=-3\end{align}\) - step9: Rewrite: \(x_{1}=-3,x_{2}=0\) Solve the equation \( 5^{x^{2}-1} \cdot \frac{1}{25}=25^{x} \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{x^{2}-1}\times \frac{1}{25}=25^{x}\) - step1: Multiply the terms: \(5^{x^{2}-3}=25^{x}\) - step2: Rewrite the expression: \(5^{x^{2}-3}=5^{2x}\) - step3: Set the exponents equal: \(x^{2}-3=2x\) - step4: Move the expression to the left side: \(x^{2}-3-2x=0\) - step5: Factor the expression: \(\left(x-3\right)\left(x+1\right)=0\) - step6: Separate into possible cases: \(\begin{align}&x-3=0\\&x+1=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=3\\&x=-1\end{align}\) - step8: Rewrite: \(x_{1}=-1,x_{2}=3\) Solve the equation \( 2^{x^{2}} \cdot 2^{x^{2}-2}+1=65 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{\left(x^{2}\right)}\times 2^{x^{2}-2}+1=65\) - step1: Multiply the terms: \(2^{2x^{2}-2}+1=65\) - step2: Move the constant to the right side: \(2^{2x^{2}-2}=65-1\) - step3: Subtract the numbers: \(2^{2x^{2}-2}=64\) - step4: Rewrite in exponential form: \(2^{2x^{2}-2}=2^{6}\) - step5: Set the exponents equal: \(2x^{2}-2=6\) - step6: Move the constant to the right side: \(2x^{2}=6+2\) - step7: Add the numbers: \(2x^{2}=8\) - step8: Divide both sides: \(\frac{2x^{2}}{2}=\frac{8}{2}\) - step9: Divide the numbers: \(x^{2}=4\) - step10: Simplify the expression: \(x=\pm \sqrt{4}\) - step11: Simplify: \(x=\pm 2\) - step12: Separate into possible cases: \(\begin{align}&x=2\\&x=-2\end{align}\) - step13: Rewrite: \(x_{1}=-2,x_{2}=2\) Solve the equation \( 10^{2 x^{2}} \cdot 10^{x+1}=10000 \). Solve the equation by following steps: - step0: Solve for \(x\): \(10^{2x^{2}}\times 10^{x+1}=10000\) - step1: Multiply: \(10^{2x^{2}+x+1}=10000\) - step2: Rewrite in exponential form: \(10^{2x^{2}+x+1}=10^{4}\) - step3: Set the exponents equal: \(2x^{2}+x+1=4\) - step4: Move the expression to the left side: \(2x^{2}+x+1-4=0\) - step5: Subtract the numbers: \(2x^{2}+x-3=0\) - step6: Factor the expression: \(\left(x-1\right)\left(2x+3\right)=0\) - step7: Separate into possible cases: \(\begin{align}&x-1=0\\&2x+3=0\end{align}\) - step8: Solve the equation: \(\begin{align}&x=1\\&x=-\frac{3}{2}\end{align}\) - step9: Rewrite: \(x_{1}=-\frac{3}{2},x_{2}=1\) Solve the equation \( 18^{x} \cdot \frac{1}{4^{x-1}}=\frac{1}{32} \). Solve the equation by following steps: - step0: Solve for \(x\): \(18^{x}\times \frac{1}{4^{x-1}}=\frac{1}{32}\) - step1: Simplify: \(9^{x}\times 2^{-x+2}=\frac{1}{32}\) - step2: Take the logarithm of both sides: \(\ln{\left(9^{x}\times 2^{-x+2}\right)}=\ln{\left(\frac{1}{32}\right)}\) - step3: Evaluate the logarithm: \(2\ln{\left(3\right)}\times x-\ln{\left(2\right)}\times x+2\ln{\left(2\right)}=\ln{\left(\frac{1}{32}\right)}\) - step4: Evaluate the logarithm: \(2\ln{\left(3\right)}\times x-\ln{\left(2\right)}\times x+2\ln{\left(2\right)}=-5\ln{\left(2\right)}\) - step5: Simplify: \(\ln{\left(\frac{3^{2}}{2}\right)}\times x+2\ln{\left(2\right)}=-5\ln{\left(2\right)}\) - step6: Move the constant to the right side: \(\ln{\left(\frac{3^{2}}{2}\right)}\times x=-5\ln{\left(2\right)}-2\ln{\left(2\right)}\) - step7: Subtract the numbers: \(\ln{\left(\frac{3^{2}}{2}\right)}\times x=-7\ln{\left(2\right)}\) - step8: Divide both sides: \(\frac{\ln{\left(\frac{3^{2}}{2}\right)}\times x}{\ln{\left(\frac{3^{2}}{2}\right)}}=\frac{-7\ln{\left(2\right)}}{\ln{\left(\frac{3^{2}}{2}\right)}}\) - step9: Divide the numbers: \(x=-7\log_{\frac{3^{2}}{2}}{\left(2\right)}\) - step10: Simplify: \(x=-7\log_{\frac{9}{2}}{\left(2\right)}\) Here are the solutions for each equation step by step: 1. **Equation:** \( 18^{x} \cdot \frac{1}{4^{x-1}}=\frac{1}{32} \) **Solution:** \[ x = -7\log_{\frac{9}{2}}{(2)} \] 2. **Equation:** \( 3^{x+1}=9^{x} \cdot \frac{1}{27} \) **Solution:** \[ x = 4 \] 3. **Equation:** \( 3^{x^{2}+2}=9 \cdot \frac{1}{27^{x}} \) **Solution:** \[ x_{1} = -3, \quad x_{2} = 0 \] 4. **Equation:** \( 5^{x^{2}-1} \cdot \frac{1}{25}=25^{x} \) **Solution:** \[ x_{1} = -1, \quad x_{2} = 3 \] 5. **Equation:** \( 2^{2 x} \cdot 4^{x+1}=1 \) **Solution:** \[ x = -\frac{1}{2} \quad \text{(or } x = -0.5\text{)} \] 6. **Equation:** \( 12^{x} \cdot 12^{x}=144 \) **Solution:** \[ x = 1 \] 7. **Equation:** \( 2^{x^{2}} \cdot 2^{x^{2}-2}+1=65 \) **Solution:** \[ x_{1} = -2, \quad x_{2} = 2 \] 8. **Equation:** \( \frac{4^{x+1} \times\left(\frac{1}{4}\right)^{-x}}{2^{2 x+1}}=\frac{1}{32} \) **Solution:** \[ x = -3 \] 9. **Equation:** \( 81 \cdot 9^{x-1}=\frac{1}{27} \) **Solution:** \[ x = -\frac{5}{2} \quad \text{(or } x = -2.5\text{)} \] 10. **Equation:** \( 10^{2 x^{2}} \cdot 10^{x+1}=10000 \) **Solution:** \[ x_{1} = -\frac{3}{2}, \quad x_{2} = 1 \quad \text{(or } x_{1} = -1.5, \quad x_{2} = 1\text{)} \] These solutions provide the values of \( x \) for each equation.