If x + `log_(10) (1 + 2^(x)) = x log_(10) 5 + log_(10)`6 then x is equal to

Options

A. 2,-3
B. 2 only
C. 1
D. 3

Answer

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

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

### Step 2: Combine logarithms
We can express \( x \log_{10} 5 + \log_{10} 6 \) as:
\[ \log_{10}(5^x) + \log_{10}(6) = \log_{10}(5^x \cdot 6) \]
Thus, we rewrite the equation:
\[ x + \log_{10}(1 + 2^x) = \log_{10}(5^x \cdot 6) \]

### Step 3: Isolate the logarithmic term
Subtract \( x \) from both sides:
\[ \log_{10}(1 + 2^x) = \log_{10}(5^x \cdot 6) - x \]

### Step 4: Rewrite \( x \) as a logarithm
We can express \( x \) as \( \log_{10}(10^x) \):
\[ \log_{10}(1 + 2^x) = \log_{10}(5^x \cdot 6) - \log_{10}(10^x) \]
Using the property of logarithms, we can combine the right side:
\[ \log_{10}(1 + 2^x) = \log_{10}\left(\frac{5^x \cdot 6}{10^x}\right) \]

### Step 5: Simplify the argument of the logarithm
This simplifies to:
\[ \log_{10}(1 + 2^x) = \log_{10}\left(6 \cdot \left(\frac{5}{10}\right)^x\right) \]
Since \(\frac{5}{10} = 0.5\), we have:
\[ \log_{10}(1 + 2^x) = \log_{10}(6 \cdot 0.5^x) \]

### Step 6: Set the arguments equal
Since the logarithms are equal, we can set the arguments equal:
\[ 1 + 2^x = 6 \cdot 0.5^x \]

### Step 7: Rewrite \( 0.5^x \)
Recall that \( 0.5^x = \frac{1}{2^x} \), so we can rewrite the equation:
\[ 1 + 2^x = \frac{6}{2^x} \]

### Step 8: Multiply through by \( 2^x \)
To eliminate the fraction, multiply both sides by \( 2^x \):
\[ 2^x + (2^x)^2 = 6 \]
This simplifies to:
\[ (2^x)^2 + 2^x - 6 = 0 \]

### Step 9: Let \( y = 2^x \)
Let \( y = 2^x \), then we have a quadratic equation:
\[ y^2 + y - 6 = 0 \]

### Step 10: Factor the quadratic
Factoring gives:
\[ (y - 2)(y + 3) = 0 \]
Thus, \( y = 2 \) or \( y = -3 \). Since \( y = 2^x \) must be positive, we discard \( y = -3 \).

### Step 11: Solve for \( x \)
Setting \( y = 2 \):
\[ 2^x = 2 \implies x = 1 \]

### Final Answer
Thus, the solution is:
\[ \boxed{1} \]