Solve for x :
`3^(x)-3^(x-1)=486`

Options

A. 7
B. 9
C. 5
D. 6

Answer

To solve the equation \( 3^x - 3^{x-1} = 486 \), we can follow these steps:

### Step 1: Rewrite the equation
We can rewrite \( 3^{x-1} \) as \( \frac{3^x}{3} \). This gives us:
\[ 3^x - \frac{3^x}{3} = 486 \]

### Step 2: Factor out \( 3^x \)
Now, we can factor \( 3^x \) from the left side:
\[ 3^x \left(1 - \frac{1}{3}\right) = 486 \]

### Step 3: Simplify the expression
The term \( 1 - \frac{1}{3} \) simplifies to \( \frac{2}{3} \):
\[ 3^x \cdot \frac{2}{3} = 486 \]

### Step 4: Multiply both sides by \( \frac{3}{2} \)
To isolate \( 3^x \), we multiply both sides by \( \frac{3}{2} \):
\[ 3^x = 486 \cdot \frac{3}{2} \]

### Step 5: Calculate the right side
Calculating \( 486 \cdot \frac{3}{2} \):
\[ 3^x = 729 \]

### Step 6: Express \( 729 \) as a power of \( 3 \)
We know that \( 729 = 3^6 \). Therefore, we can write:
\[ 3^x = 3^6 \]

### Step 7: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ x = 6 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{6} \]

---