If x + `log_(10) ( 1 + 2^(x)) = x log_(10) 5 + log_(10)6,` then the value of x is

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. 1
D. 2

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: Rearranging the equation
We start by rearranging the equation to isolate the logarithmic terms on one side:

\[ \log_{10}(1 + 2^x) - \log_{10}(6) = x \log_{10}(5) - x \]

### Step 2: Using properties of logarithms
Using the property of logarithms that states \( \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \), we can rewrite the left side:

\[ \log_{10}\left(\frac{1 + 2^x}{6}\right) = x \log_{10}(5) - x \]

### Step 3: Factoring out \( x \) on the right side
On the right side, we can factor out \( x \):

\[ \log_{10}\left(\frac{1 + 2^x}{6}\right) = x(\log_{10}(5) - 1) \]

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

\[ \frac{1 + 2^x}{6} = 10^{x(\log_{10}(5) - 1)} \]

### Step 5: Simplifying the right side
We can simplify the right side further. Since \( 10^{\log_{10}(a)} = a \), we have:

\[ \frac{1 + 2^x}{6} = \frac{5^x}{10^x} \]

### Step 6: Cross-multiplying
Cross-multiplying gives us:

\[ 1 + 2^x = 6 \cdot \frac{5^x}{10^x} \]

This simplifies to:

\[ 1 + 2^x = \frac{6 \cdot 5^x}{10^x} = \frac{6 \cdot 5^x}{(2 \cdot 5)^x} = \frac{6}{2^x} \]

### Step 7: Multiplying through by \( 2^x \)
To eliminate the fraction, we multiply through by \( 2^x \):

\[ 2^x + 2^{2x} = 6 \]

### Step 8: Rearranging into a standard form
Rearranging gives us a quadratic equation:

\[ 2^{2x} + 2^x - 6 = 0 \]

### Step 9: Substituting \( y = 2^x \)
Let \( y = 2^x \). Then we have:

\[ y^2 + y - 6 = 0 \]

### Step 10: Factoring the quadratic
Factoring the quadratic gives:

\[ (y - 2)(y + 3) = 0 \]

### Step 11: Solving for \( y \)
Setting each factor to zero gives:

\[ y - 2 = 0 \quad \Rightarrow \quad y = 2 \]
\[ y + 3 = 0 \quad \Rightarrow \quad y = -3 \quad (\text{not valid since } y = 2^x > 0) \]

### Step 12: Finding \( x \)
Since \( y = 2^x \) and \( y = 2 \):

\[ 2^x = 2 \quad \Rightarrow \quad x = 1 \]

Thus, the value of \( x \) is \( \boxed{1} \).

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