If `xlog_(10)(10//3)+log_(10)3=log_(10)(2+3^(x))+x`, then x=

Options

A. `1`
B. `0`
C. `-1`
D. `2`

Answer

To solve the equation \( x \log_{10} \left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10} (2 + 3^x) + x \), we will follow these steps:

### Step 1: Rewrite the equation using logarithmic properties
We can start by rewriting the left-hand side of the equation using the property of logarithms \( \log_a b - \log_a c = \log_a \left(\frac{b}{c}\right) \).

\[ x \log_{10} \left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10} 10 - \log_{10} 3 + \log_{10} 3 \]

This simplifies to:

\[ x \log_{10} \left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10} 10 \]

### Step 2: Isolate the logarithmic terms
Now, we can isolate the logarithmic terms on one side of the equation:

\[ x \log_{10} \left(\frac{10}{3}\right) = \log_{10} 10 - \log_{10} (2 + 3^x) \]

### Step 3: Simplify the equation
Using the property of logarithms that states \( \log_a b - \log_a c = \log_a \left(\frac{b}{c}\right) \):

\[ x \log_{10} \left(\frac{10}{3}\right) = \log_{10} \left(\frac{10}{2 + 3^x}\right) \]

### Step 4: Exponentiate both sides
To eliminate the logarithm, we exponentiate both sides:

\[ \frac{10}{3^x} = \frac{10}{2 + 3^x} \]

### Step 5: Cross-multiply
Cross-multiply to eliminate the fractions:

\[ 10(2 + 3^x) = 10 \cdot 3^x \]

### Step 6: Simplify the equation
This simplifies to:

\[ 20 + 10 \cdot 3^x = 10 \cdot 3^x \]

### Step 7: Solve for x
Subtract \( 10 \cdot 3^x \) from both sides:

\[ 20 = 0 \]

This indicates that we need to check for specific values of \( x \) that satisfy the original equation.

### Step 8: Test possible values of x
We can test \( x = 0, -1, 1, 2 \) as potential solutions:

1. **For \( x = 0 \)**:
\[ 0 \cdot \log_{10} \left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10} (2 + 3^0) + 0 \]
\[ \log_{10} 3 = \log_{10} (2 + 1) = \log_{10} 3 \quad \text{(True)} \]

2. **For \( x = -1 \)**:
\[ -1 \cdot \log_{10} \left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10} (2 + 3^{-1}) - 1 \]
This does not hold true.

3. **For \( x = 1 \)**:
\[ 1 \cdot \log_{10} \left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10} (2 + 3^1) + 1 \]
This does not hold true.

4. **For \( x = 2 \)**:
\[ 2 \cdot \log_{10} \left(\frac{10}{3}\right) + \log_{10} 3 = \log_{10} (2 + 3^2) + 2 \]
This does not hold true.

### Conclusion
The only value that satisfies the original equation is:

\[ \boxed{0} \]