To solve the equation \(2 \cdot 3^x = 3^{x+1}\), follow these steps:
1. Original Equation:
[tex]\[ 2 \cdot 3^x = 3^{x+1} \][/tex]
2. Simplify the Right-Hand Side:
Recall that \(3^{x+1}\) can be rewritten using the properties of exponents:
[tex]\[ 3^{x+1} = 3^x \cdot 3 \][/tex]
Substituting this back into the equation, we get:
[tex]\[ 2 \cdot 3^x = 3 \cdot 3^x \][/tex]
3. Isolate Terms Involving \(3^x\):
Divide both sides of the equation by \(3^x\). Since we are dividing by \(3^x\), \(3^x\) must be non-zero:
[tex]\[ 2 = 3 \][/tex]
4. Check for Possible Values:
This equation, \(2 = 3\), is clearly not true. Since there are no values of \(x\) that can make this equation true, there are no real solutions for \(x\).
Thus, the solution set for the equation \(2 \cdot 3^x = 3^{x+1}\) is empty, implying that there is no value of \(x\) that satisfies the equation.
Therefore, the correct answer from the given options is:
```
None of the provided options ( \(x=2\), \(x=-1\), \(x=0\), \(x=1\) ) are the solutions.
```
