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 start with the original equation:
\[ 3^x - 3^{x-1} = 486 \]
We can rewrite \(3^{x-1}\) as \(\frac{3^x}{3}\):
\[ 3^x - \frac{3^x}{3} = 486 \]

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

### Step 3: Isolate \(3^x\)
Next, we isolate \(3^x\) by multiplying both sides by \(\frac{3}{2}\):
\[ 3^x = 486 \cdot \frac{3}{2} \]
Calculating the right side:
\[ 3^x = \frac{1458}{2} = 729 \]

### Step 4: Express \(729\) as a power of \(3\)
We know that \(729\) can be expressed as a power of \(3\):
\[ 729 = 3^6 \]

### Step 5: 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:
\[ \boxed{6} \]