The sum of the solutions of the equations `9^(x) - 6.3^(x) +8 = 0` is

Options

A. `log_(3)2`
B. `log_(3)6`
C. `log_(3)8`
D. `log_(3)4`

Answer

To solve the equation \( 9^x - 6 \cdot 3^x + 8 = 0 \) and find the sum of the solutions, we can follow these steps:

### Step 1: Rewrite the Equation
We can express \( 9^x \) as \( (3^2)^x = 3^{2x} \). Thus, the equation becomes:
\[ 3^{2x} - 6 \cdot 3^x + 8 = 0 \]

### Step 2: Substitute \( 3^x \)
Let \( T = 3^x \). Then \( 3^{2x} = T^2 \). The equation now simplifies to:
\[ T^2 - 6T + 8 = 0 \]

### Step 3: Factor the Quadratic Equation
Next, we need to factor the quadratic equation \( T^2 - 6T + 8 = 0 \). We can look for two numbers that multiply to \( 8 \) and add up to \( -6 \). The numbers are \( -4 \) and \( -2 \). Thus, we can factor the equation as:
\[ (T - 4)(T - 2) = 0 \]

### Step 4: Solve for \( T \)
Setting each factor equal to zero gives us:
\[ T - 4 = 0 \quad \Rightarrow \quad T = 4 \]
\[ T - 2 = 0 \quad \Rightarrow \quad T = 2 \]

### Step 5: Substitute Back for \( x \)
Now, we substitute back \( T = 3^x \):
1. For \( T = 4 \):
\[ 3^x = 4 \quad \Rightarrow \quad x = \log_3(4) \]
2. For \( T = 2 \):
\[ 3^x = 2 \quad \Rightarrow \quad x = \log_3(2) \]

### Step 6: Find the Sum of the Solutions
Now, we need to find the sum of the solutions \( x_1 + x_2 \):
\[ x_1 + x_2 = \log_3(4) + \log_3(2) \]
Using the property of logarithms that states \( \log_a(b) + \log_a(c) = \log_a(bc) \), we can combine the logs:
\[ x_1 + x_2 = \log_3(4 \cdot 2) = \log_3(8) \]

### Final Answer
Thus, the sum of the solutions of the equation is:
\[ \boxed{\log_3(8)} \]