If `27^(2x-1)=(243)^(3)` then the value of x is

Options

A. 3
B. 6
C. 7
D. 9

Answer

To solve the equation \( 27^{(2x-1)} = (243)^{3} \), we will follow these steps:

### Step 1: Rewrite the bases in terms of powers of 3
We know that:
- \( 27 = 3^3 \)
- \( 243 = 3^5 \)

So we can rewrite the equation as:
\[ (3^3)^{(2x-1)} = (3^5)^{3} \]

### Step 2: Apply the power of a power property
Using the property \( (a^m)^n = a^{m \cdot n} \), we can simplify both sides:
\[ 3^{3(2x-1)} = 3^{5 \cdot 3} \]
This simplifies to:
\[ 3^{(6x - 3)} = 3^{15} \]

### Step 3: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 6x - 3 = 15 \]

### Step 4: Solve for \( x \)
Now, we will solve for \( x \):
1. Add 3 to both sides:
\[ 6x = 15 + 3 \]
\[ 6x = 18 \]
2. Divide both sides by 6:
\[ x = \frac{18}{6} \]
\[ x = 3 \]

### Final Answer
The value of \( x \) is \( 3 \).

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