The number of positive integers satisfying `x+(log)_(10)(2^x+1)=x(log)_(10)5+(log)_(10)6` is...........

Options

A. `1`
B. `6`
C. `3`
D. `4`

Explanation

To solve the equation \( x + \log_{10}(2^x + 1) = x \log_{10} 5 + \log_{10} 6 \), we will follow these steps:

### Step 1: Rewrite the Equation
Start with the given equation:
\[ x + \log_{10}(2^x + 1) = x \log_{10} 5 + \log_{10} 6 \]

### Step 2: Move \( x \) to the Right Side
Rearranging the equation gives:
\[ \log_{10}(2^x + 1) = x \log_{10} 5 + \log_{10} 6 - x \]

### Step 3: Combine Logarithmic Terms
We can express \( x \) as \( \log_{10}(10^x) \):
\[ \log_{10}(2^x + 1) = \log_{10}(5^x) + \log_{10}(6) - \log_{10}(10^x) \]
Using the property of logarithms that states \( \log a + \log b = \log(ab) \):
\[ \log_{10}(2^x + 1) = \log_{10}\left(\frac{5^x \cdot 6}{10^x}\right) \]

### Step 4: Remove the Logarithm
Since the logarithms are equal, we can set the arguments equal to each other:
\[ 2^x + 1 = \frac{5^x \cdot 6}{10^x} \]

### Step 5: Simplify the Right Side
We can rewrite \( 10^x \) as \( 2^x \cdot 5^x \):
\[ 2^x + 1 = \frac{6}{5^x} \cdot 5^x = 6 \]

### Step 6: Rearranging the Equation
This gives us:
\[ 2^x + 1 = 6 \]
Subtracting 1 from both sides results in:
\[ 2^x = 5 \]

### Step 7: Solve for \( x \)
Taking the logarithm base 2 of both sides:
\[ x = \log_{2}(5) \]

### Step 8: Check for Positive Integer Solutions
Since \( \log_{2}(5) \) is not an integer, we check if there are any positive integer solutions. The only integer solution for \( x \) can be checked by substituting small positive integers (1, 2, 3, etc.) into the original equation.

### Step 9: Substitute Values
1. For \( x = 1 \):
\[ 1 + \log_{10}(3) \approx 1 + 0.477 = 1.477 \]
\[ 1 \cdot \log_{10}(5) + \log_{10}(6) \approx 0.699 + 0.778 = 1.477 \]
Thus, \( x = 1 \) is a solution.

2. For \( x = 2 \):
\[ 2 + \log_{10}(5) \approx 2 + 0.699 = 2.699 \]
\[ 2 \cdot \log_{10}(5) + \log_{10}(6) \approx 1.398 + 0.778 = 2.176 \]
Not a solution.

3. For \( x = 3 \):
\[ 3 + \log_{10}(9) \approx 3 + 0.954 = 3.954 \]
\[ 3 \cdot \log_{10}(5) + \log_{10}(6) \approx 2.097 + 0.778 = 2.875 \]
Not a solution.

### Conclusion
The only positive integer solution is \( x = 1 \).

### Final Answer
The number of positive integers satisfying the equation is **1**.