If `3^(x-1)=9` and `3^(x+y)=81`, then value of y is:

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equations \(3^{(x-1)} = 9\) and \(3^{(x+y)} = 81\) for the value of \(y\), we can follow these steps:

### Step 1: Rewrite the equations in terms of powers of 3
We know that:
- \(9 = 3^2\)
- \(81 = 3^4\)

So we can rewrite the first equation:
\[ 3^{(x-1)} = 3^2 \]

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

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

### Step 4: Substitute \(x\) into the second equation
Now that we have \(x = 3\), we can substitute this value into the second equation:
\[ 3^{(3+y)} = 81 \]
Rewriting \(81\) as \(3^4\):
\[ 3^{(3+y)} = 3^4 \]

### Step 5: Set the exponents equal to each other again
Again, since the bases are the same, we can set the exponents equal:
\[ 3 + y = 4 \]

### Step 6: Solve for \(y\)
Now we can solve for \(y\):
\[ y = 4 - 3 = 1 \]

### Final Answer
The value of \(y\) is:
\[ \boxed{1} \]