If `(10)/(3) xx 3^(x) - 3^(x-1) = 81`, then the value of `x` is

Options

A. 2
B. 1
C. 3
D. 0

Answer

To solve the equation \(\frac{10}{3} \cdot 3^{x} - 3^{x-1} = 81\), we can follow these steps:

### Step 1: Rewrite the equation
Start with the original equation:
\[ \frac{10}{3} \cdot 3^{x} - 3^{x-1} = 81 \]

### Step 2: Simplify \(3^{x-1}\)
Recall that \(3^{x-1} = \frac{3^x}{3}\). Substitute this into the equation:
\[ \frac{10}{3} \cdot 3^{x} - \frac{3^{x}}{3} = 81 \]

### Step 3: Combine the terms
Now, factor out \(3^{x}\) from the left side:
\[ \frac{10 \cdot 3^{x}}{3} - \frac{3^{x}}{3} = 81 \]
This can be rewritten as:
\[ \frac{3^{x}}{3} (10 - 1) = 81 \]
Thus, we have:
\[ \frac{9 \cdot 3^{x}}{3} = 81 \]

### Step 4: Simplify further
Now simplify:
\[ 3^{x} \cdot 3 = 81 \]
This simplifies to:
\[ 3^{x+1} = 81 \]

### Step 5: Express 81 as a power of 3
Recognize that \(81\) can be expressed as \(3^{4}\):
\[ 3^{x+1} = 3^{4} \]

### Step 6: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ x + 1 = 4 \]

### Step 7: Solve for \(x\)
Now, solve for \(x\):
\[ x = 4 - 1 = 3 \]

### Final Answer
The value of \(x\) is:
\[ \boxed{3} \]

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