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

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 by rewriting the equation for clarity:
\[ \frac{10}{3} \cdot 3^x - 3^{x-1} = 81 \]

### Step 2: Factor out \(3^{x-1}\)
Notice that \(3^{x-1}\) can be factored out from both terms. Recall that \(3^{x-1} = \frac{3^x}{3}\). Thus, we can rewrite the equation as:
\[ \frac{10}{3} \cdot 3^x - \frac{3^x}{3} = 81 \]
This simplifies to:
\[ \frac{3^x}{3} \left(10 - 1\right) = 81 \]
or
\[ 3^{x-1} \cdot 9 = 81 \]

### Step 3: Simplify the equation
Now, divide both sides by 9:
\[ 3^{x-1} = \frac{81}{9} \]
This simplifies to:
\[ 3^{x-1} = 9 \]

### Step 4: Express 9 as a power of 3
We know that \(9\) can be expressed as \(3^2\):
\[ 3^{x-1} = 3^2 \]

### Step 5: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:
\[ x - 1 = 2 \]

### Step 6: Solve for \(x\)
Now, solve for \(x\):
\[ x = 2 + 1 = 3 \]

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

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