\( \begin{array}{lll}\text { (e) }-2^{x}=-16 & \text { (f) }-2^{x}=-32 \\ \text { (h) }-2^{x}=-1 & \text { (i) } 7^{x}=49 \\ \text { (k) } 7^{x}=1 & \text { (l) } 3^{2 x}=81 \\ \text { (n) } 10^{10 x}=100 & \text { (o) } 4^{x}=2 \\ \text { (q) } 25^{x}=125 & \text { (r) } 9^{3 x}=27^{x} \\ \text { (t) } 9^{3 x-1}=27 & \text { (u) } 16^{2 x}=4^{x-1}\end{array} \)

Solve the equation by following steps: - step0: Solve for \(x\): \(3^{2x}=81\) - step1: Rewrite in exponential form: \(3^{2x}=3^{4}\) - step2: Set the exponents equal: \(2x=4\) - step3: Divide both sides: \(\frac{2x}{2}=\frac{4}{2}\) - step4: Divide the numbers: \(x=2\) Solve the equation \( 16^{2 x}=4^{x-1} \). Solve the equation by following steps: - step0: Solve for \(x\): \(16^{2x}=4^{x-1}\) - step1: Rewrite the expression: \(4^{4x}=4^{x-1}\) - step2: Set the exponents equal: \(4x=x-1\) - step3: Move the variable to the left side: \(4x-x=-1\) - step4: Subtract the terms: \(3x=-1\) - step5: Divide both sides: \(\frac{3x}{3}=\frac{-1}{3}\) - step6: Divide the numbers: \(x=-\frac{1}{3}\) Solve the equation \( 25^{x}=125 \). Solve the equation by following steps: - step0: Solve for \(x\): \(25^{x}=125\) - step1: Rewrite in exponential form: \(5^{2x}=5^{3}\) - step2: Set the exponents equal: \(2x=3\) - step3: Divide both sides: \(\frac{2x}{2}=\frac{3}{2}\) - step4: Divide the numbers: \(x=\frac{3}{2}\) Solve the equation \( -2^{x}=-1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(-2^{x}=-1\) - step1: Change the signs: \(2^{x}=1\) - step2: Rewrite in exponential form: \(2^{x}=2^{0}\) - step3: Set the exponents equal: \(x=0\) Solve the equation \( 10^{10 x}=100 \). Solve the equation by following steps: - step0: Solve for \(x\): \(10^{10x}=100\) - step1: Rewrite in exponential form: \(10^{10x}=10^{2}\) - step2: Set the exponents equal: \(10x=2\) - step3: Divide both sides: \(\frac{10x}{10}=\frac{2}{10}\) - step4: Divide the numbers: \(x=\frac{1}{5}\) Solve the equation \( -2^{x}=-16 \). Solve the equation by following steps: - step0: Solve for \(x\): \(-2^{x}=-16\) - step1: Change the signs: \(2^{x}=16\) - step2: Rewrite in exponential form: \(2^{x}=2^{4}\) - step3: Set the exponents equal: \(x=4\) Solve the equation \( -2^{x}=-32 \). Solve the equation by following steps: - step0: Solve for \(x\): \(-2^{x}=-32\) - step1: Change the signs: \(2^{x}=32\) - step2: Rewrite in exponential form: \(2^{x}=2^{5}\) - step3: Set the exponents equal: \(x=5\) Solve the equation \( 7^{x}=49 \). Solve the equation by following steps: - step0: Solve for \(x\): \(7^{x}=49\) - step1: Rewrite in exponential form: \(7^{x}=7^{2}\) - step2: Set the exponents equal: \(x=2\) Solve the equation \( 7^{x}=1 \). Solve the equation by following steps: - step0: Solve for \(x\): \(7^{x}=1\) - step1: Rewrite in exponential form: \(7^{x}=7^{0}\) - step2: Set the exponents equal: \(x=0\) Solve the equation \( 4^{x}=2 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4^{x}=2\) - step1: Rewrite in exponential form: \(2^{2x}=2\) - step2: Set the exponents equal: \(2x=1\) - step3: Divide both sides: \(\frac{2x}{2}=\frac{1}{2}\) - step4: Divide the numbers: \(x=\frac{1}{2}\) Solve the equation \( 9^{3 x}=27^{x} \). Solve the equation by following steps: - step0: Solve for \(x\): \(9^{3x}=27^{x}\) - step1: Rewrite the expression: \(3^{6x}=3^{3x}\) - step2: Set the exponents equal: \(6x=3x\) - step3: Add or subtract both sides: \(6x-3x=0\) - step4: Subtract the terms: \(3x=0\) - step5: Rewrite the expression: \(x=0\) Solve the equation \( 9^{3 x-1}=27 \). Solve the equation by following steps: - step0: Solve for \(x\): \(9^{3x-1}=27\) - step1: Rewrite in exponential form: \(3^{2\left(3x-1\right)}=3^{3}\) - step2: Set the exponents equal: \(2\left(3x-1\right)=3\) - step3: Divide both sides: \(\frac{2\left(3x-1\right)}{2}=\frac{3}{2}\) - step4: Divide the numbers: \(3x-1=\frac{3}{2}\) - step5: Move the constant to the right side: \(3x=\frac{3}{2}+1\) - step6: Add the numbers: \(3x=\frac{5}{2}\) - step7: Multiply by the reciprocal: \(3x\times \frac{1}{3}=\frac{5}{2}\times \frac{1}{3}\) - step8: Multiply: \(x=\frac{5}{6}\) Here are the solutions for each equation: (e) \( -2^{x} = -16 \) To solve for \( x \): \[ -2^{x} = -16 \implies 2^{x} = 16 \implies 2^{x} = 2^{4} \implies x = 4 \] **Solution:** \( x = 4 \) (f) \( -2^{x} = -32 \) To solve for \( x \): \[ -2^{x} = -32 \implies 2^{x} = 32 \implies 2^{x} = 2^{5} \implies x = 5 \] **Solution:** \( x = 5 \) (h) \( -2^{x} = -1 \) To solve for \( x \): \[ -2^{x} = -1 \implies 2^{x} = 1 \implies 2^{x} = 2^{0} \implies x = 0 \] **Solution:** \( x = 0 \) (i) \( 7^{x} = 49 \) To solve for \( x \): \[ 7^{x} = 49 \implies 7^{x} = 7^{2} \implies x = 2 \] **Solution:** \( x = 2 \) (k) \( 7^{x} = 1 \) To solve for \( x \): \[ 7^{x} = 1 \implies 7^{x} = 7^{0} \implies x = 0 \] **Solution:** \( x = 0 \) (l) \( 3^{2x} = 81 \) To solve for \( x \): \[ 3^{2x} = 81 \implies 3^{2x} = 3^{4} \implies 2x = 4 \implies x = 2 \] **Solution:** \( x = 2 \) (n) \( 10^{10x} = 100 \) To solve for \( x \): \[ 10^{10x} = 100 \implies 10^{10x} = 10^{2} \implies 10x = 2 \implies x = \frac{1}{5} \text{ or } 0.2 \] **Solution:** \( x = \frac{1}{5} \) (o) \( 4^{x} = 2 \) To solve for \( x \): \[ 4^{x} = 2 \implies (2^{2})^{x} = 2 \implies 2^{2x} = 2^{1} \implies 2x = 1 \implies x = \frac{1}{2} \text{ or } 0.5 \] **Solution:** \( x = \frac{1}{2} \) (q) \( 25^{x} = 125 \) To solve for \( x \): \[ 25^{x} = 125 \implies (5^{2})^{x} = 5^{3} \implies 5^{2x} = 5^{3} \implies 2x = 3 \implies x = \frac{3}{2} \text{ or } 1.5 \] **Solution:** \( x = \frac{3}{2} \) (r) \( 9^{3x} = 27^{x} \) To solve for \( x \): \[ 9^{3x} = 27^{x} \implies (3^{2})^{3x} = (3^{3})^{x} \implies 3^{6x} = 3^{3x} \implies 6x = 3x \implies 3x = 0 \implies x = 0 \] **Solution:** \( x = 0 \) (t) \( 9^{3x-1} = 27 \) To solve for \( x \): \[ 9^{3x-1} = 27 \implies (3^{2})^{3x-1} = 3^{3} \implies 3^{6x-2} = 3^{3} \implies 6x - 2 = 3 \implies 6x = 5 \implies x = \frac{5}{6} \text{ or } 0.8333 \] **Solution:** \( x = \frac{5}{6} \) (u) \( 16^{2x} = 4^{x-1} \) To solve for \( x \): \[ 16^{2x} = 4^{x-1} \implies (4^{2})^{2x} = 4^{x-1} \implies 4^{4x} = 4^{x-1} \implies 4x = x - 1 \implies 3x = -1 \implies x = -\frac{1}{3} \text{ or } -0.3333 \] **Solution:** \( x = -\frac{1}{3} \) These are the solutions for all the equations provided.