\( \left. \begin{array} { l l } { 2 } & { 3 ^ { x + 1 } = 3 ^ { x } + \frac { 2 } { 3 } } \\ { 3 } & { 3 ^ { x } + 3 ^ { x + 2 } = 10 } \\ { 4 } & { 5 ^ { x + 1 } - \frac { 3 } { 25 } = 2.5 ^ { x } } \\ { 5 } & { 2 ^ { x } + 2 ^ { x } + 2 ^ { x } = 96 } \\ { 6 } & { 12 ^ { x } + 12 ^ { x } = 288 } \\ { 7 } & { 2 ^ { x } + 2 ^ { x - 2 } = 5 } \end{array} \right. \)

Sure, let's solve each of the equations step by step. --- ### **Problem 2** \[ 3^{x+1} = 3^{x} + \frac{2}{3} \] **Solution:** 1. **Simplify the left side:** \[ 3^{x+1} = 3^x \cdot 3 = 3 \cdot 3^x \] 2. **Rewrite the equation:** \[ 3 \cdot 3^x = 3^x + \frac{2}{3} \] 3. **Subtract \(3^x\) from both sides:** \[ 2 \cdot 3^x = \frac{2}{3} \] 4. **Divide both sides by 2:** \[ 3^x = \frac{1}{3} \] 5. **Express \(\frac{1}{3}\) as \(3^{-1}\):** \[ 3^x = 3^{-1} \] 6. **Since the bases are equal, set the exponents equal:** \[ x = -1 \] **Answer:** \( x = -1 \) --- ### **Problem 3** \[ 3^{x} + 3^{x+2} = 10 \] **Solution:** 1. **Factor out \(3^x\):** \[ 3^x (1 + 3^2) = 10 \] 2. **Calculate \(3^2\):** \[ 3^x (1 + 9) = 10 \Rightarrow 3^x \cdot 10 = 10 \] 3. **Divide both sides by 10:** \[ 3^x = 1 \] 4. **Express 1 as \(3^0\):** \[ 3^x = 3^0 \] 5. **Set the exponents equal:** \[ x = 0 \] **Answer:** \( x = 0 \) --- ### **Problem 4** \[ 5^{x+1} - \frac{3}{25} = 2.5^{x} \] **Solution:** 1. **Express 2.5 in terms of base 5:** \[ 2.5 = \frac{5}{2} \Rightarrow 2.5^x = \left(\frac{5}{2}\right)^x \] 2. **Express \(\frac{5}{2}\) as \(5^x \cdot 2^{-x}\):** \[ 5^{x+1} - \frac{3}{25} = 5^x \cdot 2^{-x} \] 3. **Divide both sides by \(5^x\):** \[ 5 \cdot 5^x \cdot 5^{-x} - \frac{3}{25} \cdot 5^{-x} = 2^{-x} \] \[ 5 - \frac{3}{25} \cdot 5^{-x} = 2^{-x} \] This equation is transcendental and doesn't have an elementary algebraic solution. Therefore, we can use numerical methods or graphing to approximate the solution. 4. **Using numerical methods (e.g., trial and error):** Let's test \(x = 1\): \[ 5^{2} - \frac{3}{25} = 25 - 0.12 = 24.88 \] \[ 2.5^{1} = 2.5 \] Not equal. Try \(x = 2\): \[ 5^{3} - \frac{3}{25} = 125 - 0.12 = 124.88 \] \[ 2.5^{2} = 6.25 \] Still not equal. Since the left side grows much faster than the right side, there is **no real solution**. **Answer:** No real solution exists for this equation. --- ### **Problem 5** \[ 2^{x} + 2^{x} + 2^{x} = 96 \] **Solution:** 1. **Combine like terms:** \[ 3 \cdot 2^x = 96 \] 2. **Divide both sides by 3:** \[ 2^x = 32 \] 3. **Express 32 as a power of 2:** \[ 32 = 2^5 \] 4. **Set the exponents equal:** \[ x = 5 \] **Answer:** \( x = 5 \) --- ### **Problem 6** \[ 12^{x} + 12^{x} = 288 \] **Solution:** 1. **Combine like terms:** \[ 2 \cdot 12^x = 288 \] 2. **Divide both sides by 2:** \[ 12^x = 144 \] 3. **Express 144 as a power of 12:** \[ 12^2 = 144 \] 4. **Set the exponents equal:** \[ x = 2 \] **Answer:** \( x = 2 \) --- ### **Problem 7** \[ 2^{x} + 2^{x-2} = 5 \] **Solution:** 1. **Express \(2^{x-2}\) in terms of \(2^x\):** \[ 2^{x-2} = \frac{2^x}{2^2} = \frac{2^x}{4} \] 2. **Rewrite the equation:** \[ 2^x + \frac{2^x}{4} = 5 \] 3. **Combine like terms:** \[ \frac{4 \cdot 2^x + 2^x}{4} = 5 \] \[ \frac{5 \cdot 2^x}{4} = 5 \] 4. **Multiply both sides by 4:** \[ 5 \cdot 2^x = 20 \] 5. **Divide both sides by 5:** \[ 2^x = 4 \] 6. **Express 4 as a power of 2:** \[ 4 = 2^2 \] 7. **Set the exponents equal:** \[ x = 2 \] **Answer:** \( x = 2 \) --- If you have any further questions or need additional explanations, feel free to ask!