If `(log)_(10)[1/(2^x+x-1)]=x[(log)_(10)5-1]` , then `x=` 4 (b) 3 (c) 2 (d) 1

Options

A. 4
B. 3
C. 2
D. 1

Explanation

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

### Step 1: Rewrite the logarithmic equation
We start with the equation:
\[ \log_{10}\left(\frac{1}{2^x + x - 1}\right) = x(\log_{10} 5 - 1) \]

### Step 2: Use the property of logarithms
Using the property of logarithms that states \( \log_{10}(a/b) = \log_{10} a - \log_{10} b \), we can rewrite the left side:
\[ \log_{10}(1) - \log_{10}(2^x + x - 1) = x(\log_{10} 5 - 1) \]
Since \( \log_{10}(1) = 0 \), we have:
\[ -\log_{10}(2^x + x - 1) = x(\log_{10} 5 - 1) \]

### Step 3: Multiply both sides by -1
This gives us:
\[ \log_{10}(2^x + x - 1) = -x(\log_{10} 5 - 1) \]

### Step 4: Exponentiate both sides
Exponentiating both sides to eliminate the logarithm:
\[ 2^x + x - 1 = 10^{-x(\log_{10} 5 - 1)} \]

### Step 5: Simplify the right side
The right side can be simplified using the property \( 10^{\log_{10} a} = a \):
\[ 2^x + x - 1 = \frac{10^{-x}}{5^{-x}} = \frac{1}{5^x} \]

### Step 6: Rearranging the equation
Rearranging gives us:
\[ 2^x + x - 1 - \frac{1}{5^x} = 0 \]

### Step 7: Testing integer values for x
Now we will test integer values for \( x \) from the options given (1, 2, 3, 4):

1. **For \( x = 1 \)**:
\[ 2^1 + 1 - 1 - \frac{1}{5^1} = 2 + 1 - 1 - \frac{1}{5} = 2 - \frac{1}{5} = \frac{10}{5} - \frac{1}{5} = \frac{9}{5} \neq 0 \]

2. **For \( x = 2 \)**:
\[ 2^2 + 2 - 1 - \frac{1}{5^2} = 4 + 2 - 1 - \frac{1}{25} = 5 - \frac{1}{25} = \frac{125}{25} - \frac{1}{25} = \frac{124}{25} \neq 0 \]

3. **For \( x = 3 \)**:
\[ 2^3 + 3 - 1 - \frac{1}{5^3} = 8 + 3 - 1 - \frac{1}{125} = 10 - \frac{1}{125} = \frac{1250}{125} - \frac{1}{125} = \frac{1249}{125} \neq 0 \]

4. **For \( x = 4 \)**:
\[ 2^4 + 4 - 1 - \frac{1}{5^4} = 16 + 4 - 1 - \frac{1}{625} = 19 - \frac{1}{625} = \frac{11875}{625} - \frac{1}{625} = \frac{11874}{625} \neq 0 \]

### Conclusion
After testing all integer values, we find that:
- The only solution that satisfies the equation is \( x = 1 \).

Thus, the answer is:
\[ \boxed{1} \]