2. Solve for \( x \) : \( \begin{array}{ll}\text { a) } 2^{2 x-1}=8 & \text { b) } 3^{\frac{3 x-1}{2}}=\frac{1}{27} \\ \text { c) } 4^{x-1}=32 & \text { d) } 9^{2 x+1}=\frac{1}{27} \\ \text { e) } 5^{-2}=\frac{1}{25^{x}} & \text { (f) } 3^{x-2}=\frac{1}{3^{x}} \\ \text { g) } 4.16^{x}=8 & \text { h) } 2.5^{x}=50 \\ \text { i) } 1-6^{x}=0 & \text { j) } 10^{x}=1 \\ \text { k) } 3^{x-1}=1 & \text { (l) } 4^{-x}-16=0 \\ \text { m) } 4.5^{2 x}=4 & \text { b) } 8^{x+2}=16^{2 x-3} \\ \text { 3. Solve for } x: & \text { d) } 5^{2 x+3}=0,008 \\ \text { a) } 9^{2 x+3}=27^{x+5} & \text { f) } 5^{x(x+1)}=25 \\ \text { c) } 27^{x-2}=81^{2 x+1} & \\ \text { e) } 2^{x+1}=(0,5)^{x-2} & \\ \text { g) } 5^{2 x(x-3)}=0,20 & \end{array} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(10^{x}=1\) - step1: Rewrite in exponential form: \(10^{x}=10^{0}\) - step2: Set the exponents equal: \(x=0\) Solve the equation \( 1-6^{x}=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(1-6^{x}=0\) - step1: Move the constant to the right side: \(-6^{x}=0-1\) - step2: Remove 0: \(-6^{x}=-1\) - step3: Change the signs: \(6^{x}=1\) - step4: Rewrite in exponential form: \(6^{x}=6^{0}\) - step5: Set the exponents equal: \(x=0\) Solve the equation \( 2^{2 x-1}=8 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{2x-1}=8\) - step1: Rewrite in exponential form: \(2^{2x-1}=2^{3}\) - step2: Set the exponents equal: \(2x-1=3\) - step3: Move the constant to the right side: \(2x=3+1\) - step4: Add the numbers: \(2x=4\) - step5: Divide both sides: \(\frac{2x}{2}=\frac{4}{2}\) - step6: Divide the numbers: \(x=2\) Solve the equation \( 9^{2 x+3}=27^{x+5} \). Solve the equation by following steps: - step0: Solve for \(x\): \(9^{2x+3}=27^{x+5}\) - step1: Rewrite the expression: \(3^{4x+6}=3^{3x+15}\) - step2: Set the exponents equal: \(4x+6=3x+15\) - step3: Move the expression to the left side: \(4x-3x=15-6\) - step4: Add and subtract: \(x=15-6\) - step5: Add and subtract: \(x=9\) Solve the equation \( 4^{x-1}=32 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4^{x-1}=32\) - step1: Rewrite in exponential form: \(2^{2\left(x-1\right)}=2^{5}\) - step2: Set the exponents equal: \(2\left(x-1\right)=5\) - step3: Divide both sides: \(\frac{2\left(x-1\right)}{2}=\frac{5}{2}\) - step4: Divide the numbers: \(x-1=\frac{5}{2}\) - step5: Move the constant to the right side: \(x=\frac{5}{2}+1\) - step6: Add the numbers: \(x=\frac{7}{2}\) Solve the equation \( 27^{x-2}=81^{2 x+1} \). Solve the equation by following steps: - step0: Solve for \(x\): \(27^{x-2}=81^{2x+1}\) - step1: Rewrite the expression: \(3^{3x-6}=3^{8x+4}\) - step2: Set the exponents equal: \(3x-6=8x+4\) - step3: Move the expression to the left side: \(3x-8x=4+6\) - step4: Add and subtract: \(-5x=4+6\) - step5: Add and subtract: \(-5x=10\) - step6: Change the signs: \(5x=-10\) - step7: Divide both sides: \(\frac{5x}{5}=\frac{-10}{5}\) - step8: Divide the numbers: \(x=-2\) Solve the equation \( 4.16^{x}=8 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4.16^{x}=8\) - step1: Convert the expressions: \(\left(\frac{104}{25}\right)^{x}=8\) - step2: Take the logarithm of both sides: \(\log_{\frac{104}{25}}{\left(\left(\frac{104}{25}\right)^{x}\right)}=\log_{\frac{104}{25}}{\left(8\right)}\) - step3: Evaluate the logarithm: \(x=\log_{\frac{104}{25}}{\left(8\right)}\) - step4: Simplify: \(x=3\log_{\frac{104}{25}}{\left(2\right)}\) Solve the equation \( 3^{x-2}=1/(3^{x}) \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x-2}=\frac{1}{3^{x}}\) - step1: Rewrite the expression: \(3^{x-2}=3^{-x}\) - step2: Set the exponents equal: \(x-2=-x\) - step3: Move the variable to the left side: \(x-2+x=0\) - step4: Add the terms: \(2x-2=0\) - step5: Move the constant to the right side: \(2x=0+2\) - step6: Remove 0: \(2x=2\) - step7: Divide both sides: \(\frac{2x}{2}=\frac{2}{2}\) - step8: Divide the numbers: \(x=1\) Solve the equation \( 4^{-x}-16=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4^{-x}-16=0\) - step1: Move the constant to the right side: \(4^{-x}=0+16\) - step2: Remove 0: \(4^{-x}=16\) - step3: Rewrite in exponential form: \(4^{-x}=4^{2}\) - step4: Set the exponents equal: \(-x=2\) - step5: Change the signs: \(x=-2\) Solve the equation \( 2^{x+1}=(0.5)^{x-2} \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{x+1}=0.5^{x-2}\) - step1: Convert the expressions: \(2^{x+1}=\left(\frac{1}{2}\right)^{x-2}\) - step2: Rewrite the expression: \(2^{x+1}=2^{-x+2}\) - step3: Set the exponents equal: \(x+1=-x+2\) - step4: Move the expression to the left side: \(x+x=2-1\) - step5: Add and subtract: \(2x=2-1\) - step6: Add and subtract: \(2x=1\) - step7: Divide both sides: \(\frac{2x}{2}=\frac{1}{2}\) - step8: Divide the numbers: \(x=\frac{1}{2}\) Solve the equation \( 9^{2 x+1}=1/27 \). Solve the equation by following steps: - step0: Solve for \(x\): \(9^{2x+1}=\frac{1}{27}\) - step1: Rewrite in exponential form: \(3^{2\left(2x+1\right)}=3^{-3}\) - step2: Set the exponents equal: \(2\left(2x+1\right)=-3\) - step3: Divide both sides: \(\frac{2\left(2x+1\right)}{2}=\frac{-3}{2}\) - step4: Divide the numbers: \(2x+1=-\frac{3}{2}\) - step5: Move the constant to the right side: \(2x=-\frac{3}{2}-1\) - step6: Subtract the numbers: \(2x=-\frac{5}{2}\) - step7: Multiply by the reciprocal: \(2x\times \frac{1}{2}=-\frac{5}{2}\times \frac{1}{2}\) - step8: Multiply: \(x=-\frac{5}{4}\) Solve the equation \( 5^{2 x+3}=0.008 \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{2x+3}=0.008\) - step1: Rewrite the expression: \(5^{2x+3}=\frac{1}{125}\) - step2: Rewrite the expression: \(5^{2x+3}=5^{-3}\) - step3: Set the exponents equal: \(2x+3=-3\) - step4: Move the constant to the right side: \(2x=-3-3\) - 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 \( 8^{x+2}=16^{2 x-3} \). Solve the equation by following steps: - step0: Solve for \(x\): \(8^{x+2}=16^{2x-3}\) - step1: Rewrite the expression: \(2^{3x+6}=2^{8x-12}\) - step2: Set the exponents equal: \(3x+6=8x-12\) - step3: Move the expression to the left side: \(3x-8x=-12-6\) - step4: Add and subtract: \(-5x=-12-6\) - step5: Add and subtract: \(-5x=-18\) - step6: Change the signs: \(5x=18\) - step7: Divide both sides: \(\frac{5x}{5}=\frac{18}{5}\) - step8: Divide the numbers: \(x=\frac{18}{5}\) Solve the equation \( 3^{x-1}=1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{x-1}=1\) - step1: Rewrite in exponential form: \(3^{x-1}=3^{0}\) - step2: Set the exponents equal: \(x-1=0\) - step3: Move the constant to the right side: \(x=0+1\) - step4: Remove 0: \(x=1\) Solve the equation \( 2.5^{x}=50 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2.5^{x}=50\) - step1: Convert the expressions: \(\left(\frac{5}{2}\right)^{x}=50\) - step2: Take the logarithm of both sides: \(\log_{\frac{5}{2}}{\left(\left(\frac{5}{2}\right)^{x}\right)}=\log_{\frac{5}{2}}{\left(50\right)}\) - step3: Evaluate the logarithm: \(x=\log_{\frac{5}{2}}{\left(50\right)}\) Solve the equation \( 5^{-2}=1/(25^{x}) \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{-2}=\frac{1}{\left(25^{x}\right)}\) - step1: Evaluate: \(5^{-2}=\frac{1}{25^{x}}\) - step2: Rearrange the terms: \(5^{-2}=25^{-x}\) - step3: Swap the sides: \(25^{-x}=5^{-2}\) - step4: Rewrite in exponential form: \(5^{-2x}=5^{-2}\) - step5: Set the exponents equal: \(-2x=-2\) - step6: Change the signs: \(2x=2\) - step7: Divide both sides: \(\frac{2x}{2}=\frac{2}{2}\) - step8: Divide the numbers: \(x=1\) Solve the equation \( 3^{(3 x-1)/2}=1/27 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{\frac{\left(3x-1\right)}{2}}=\frac{1}{27}\) - step1: Remove the parentheses: \(3^{\frac{3x-1}{2}}=\frac{1}{27}\) - step2: Rewrite in exponential form: \(3^{\frac{3x-1}{2}}=3^{-3}\) - step3: Set the exponents equal: \(\frac{3x-1}{2}=-3\) - step4: Cross multiply: \(3x-1=2\left(-3\right)\) - step5: Simplify the equation: \(3x-1=-6\) - step6: Move the constant to the right side: \(3x=-6+1\) - step7: Add the numbers: \(3x=-5\) - step8: Divide both sides: \(\frac{3x}{3}=\frac{-5}{3}\) - step9: Divide the numbers: \(x=-\frac{5}{3}\) Solve the equation \( 4.5^{2 x}=4 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4.5^{2x}=4\) - step1: Convert the expressions: \(\left(\frac{9}{2}\right)^{2x}=4\) - step2: Take the logarithm of both sides: \(\log_{\frac{9}{2}}{\left(\left(\frac{9}{2}\right)^{2x}\right)}=\log_{\frac{9}{2}}{\left(4\right)}\) - step3: Evaluate the logarithm: \(2x=\log_{\frac{9}{2}}{\left(4\right)}\) - step4: Divide both sides: \(\frac{2x}{2}=\frac{\log_{\frac{9}{2}}{\left(4\right)}}{2}\) - step5: Divide the numbers: \(x=\frac{\log_{\frac{9}{2}}{\left(4\right)}}{2}\) - step6: Simplify: \(x=\log_{\frac{9}{2}}{\left(2\right)}\) Solve the equation \( 5^{x(x+1)}=25 \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{x\left(x+1\right)}=25\) - step1: Rewrite in exponential form: \(5^{x\left(x+1\right)}=5^{2}\) - step2: Set the exponents equal: \(x\left(x+1\right)=2\) - step3: Expand the expression: \(x^{2}+x=2\) - step4: Move the expression to the left side: \(x^{2}+x-2=0\) - step5: Factor the expression: \(\left(x-1\right)\left(x+2\right)=0\) - step6: Separate into possible cases: \(\begin{align}&x-1=0\\&x+2=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=1\\&x=-2\end{align}\) - step8: Rewrite: \(x_{1}=-2,x_{2}=1\) Solve the equation \( 5^{2 x(x-3)}=0.20 \). Solve the equation by following steps: - step0: Solve for \(x\): \(5^{2x\left(x-3\right)}=0.2\) - step1: Rewrite in exponential form: \(5^{2x\left(x-3\right)}=5^{-1}\) - step2: Set the exponents equal: \(2x\left(x-3\right)=-1\) - step3: Expand the expression: \(2x^{2}-6x=-1\) - step4: Move the expression to the left side: \(2x^{2}-6x-\left(-1\right)=0\) - step5: Remove the parentheses: \(2x^{2}-6x+1=0\) - step6: Solve using the quadratic formula: \(x=\frac{6\pm \sqrt{\left(-6\right)^{2}-4\times 2}}{2\times 2}\) - step7: Simplify the expression: \(x=\frac{6\pm \sqrt{\left(-6\right)^{2}-4\times 2}}{4}\) - step8: Simplify the expression: \(x=\frac{6\pm \sqrt{28}}{4}\) - step9: Simplify the expression: \(x=\frac{6\pm 2\sqrt{7}}{4}\) - step10: Separate into possible cases: \(\begin{align}&x=\frac{6+2\sqrt{7}}{4}\\&x=\frac{6-2\sqrt{7}}{4}\end{align}\) - step11: Simplify the expression: \(\begin{align}&x=\frac{3+\sqrt{7}}{2}\\&x=\frac{6-2\sqrt{7}}{4}\end{align}\) - step12: Simplify the expression: \(\begin{align}&x=\frac{3+\sqrt{7}}{2}\\&x=\frac{3-\sqrt{7}}{2}\end{align}\) - step13: Rewrite: \(x_{1}=\frac{3-\sqrt{7}}{2},x_{2}=\frac{3+\sqrt{7}}{2}\) Here are the solutions for each equation: ### 1. Solve for \( x \): **a)** \( 2^{2x-1} = 8 \) Solution: \( x = 2 \) **b)** \( 3^{\frac{3x-1}{2}} = \frac{1}{27} \) Solution: \( x = -\frac{5}{3} \) or approximately \( x \approx -1.67 \) **c)** \( 4^{x-1} = 32 \) Solution: \( x = \frac{7}{2} \) or \( x = 3.5 \) **d)** \( 9^{2x+1} = \frac{1}{27} \) Solution: \( x = -\frac{5}{4} \) or \( x = -1.25 \) **e)** \( 5^{-2} = \frac{1}{25^{x}} \) Solution: \( x = 1 \) **f)** \( 3^{x-2} = \frac{1}{3^{x}} \) Solution: \( x = 1 \) **g)** \( 4.16^{x} = 8 \) Solution: \( x = 3 \log_{\frac{104}{25}}{(2)} \) **h)** \( 2.5^{x} = 50 \) Solution: \( x = \log_{\frac{5}{2}}{(50)} \) **i)** \( 1 - 6^{x} = 0 \) Solution: \( x = 0 \) **j)** \( 10^{x} = 1 \) Solution: \( x = 0 \) **k)** \( 3^{x-1} = 1 \) Solution: \( x = 1 \) **l)** \( 4^{-x} - 16 = 0 \) Solution: \( x = -2 \) **m)** \( 4.5^{2x} = 4 \) Solution: \( x = \log_{\frac{9}{2}}{(2)} \) **n)** \( 8^{x+2} = 16^{2x-3} \) Solution: \( x = \frac{18}{5} \) or \( x = 3.6 \) ### 2. Solve for \( x \): **a)** \( 5^{2x+3} = 0.008 \) Solution: \( x = -3 \) **b)** \( 9^{2x+3} = 27^{x+5} \) Solution: \( x_{1} = -2, x_{2} = 1 \) **c)** \( 5^{x(x+1)} = 25 \) Solution: \( x = -2 \) **d)** \( 27^{x-2} = 81^{2x+1} \) Solution: \( x = \frac{1}{2} \) or \( x = 0.5 \) **e)** \( 2^{x+1} = (0.5)^{x-2} \) Solution: \( x_{1} = \frac{3 - \sqrt{7}}{2}, x_{2} = \frac{3 + \sqrt{7}}{2} \) **f)** \( 5^{2x(x-3)} = 0.20 \) Solution: \( x = -2 \) These solutions cover all the equations provided. If you need further clarification on any specific solution, feel free to ask!