The sum of all the values of a satisfying the equation `|[log_10 a,-1],[log_10(a-1),2]|=log_10 a+log_10 2`

Options

A. 0
B. 1
C. 2
D. none of these

Explanation

To solve the equation \( |[\log_{10} a, -1], [\log_{10}(a-1), 2]| = \log_{10} a + \log_{10} 2 \), we will follow these steps:

### Step 1: Calculate the Determinant
The determinant of the matrix \( |[\log_{10} a, -1], [\log_{10}(a-1), 2]| \) can be calculated using the formula for the determinant of a 2x2 matrix:
\[ \text{det} = ad - bc \]
where \( a = \log_{10} a \), \( b = -1 \), \( c = \log_{10}(a-1) \), and \( d = 2 \).

Thus, we have:
\[ \text{det} = (\log_{10} a)(2) - (-1)(\log_{10}(a-1)) = 2 \log_{10} a + \log_{10}(a-1) \]

### Step 2: Set the Determinant Equal to the Right Side
Now we set the determinant equal to the right side of the equation:
\[ 2 \log_{10} a + \log_{10}(a-1) = \log_{10} a + \log_{10} 2 \]

### Step 3: Simplify the Equation
We can simplify the equation:
\[ 2 \log_{10} a + \log_{10}(a-1) - \log_{10} a - \log_{10} 2 = 0 \]
This simplifies to:
\[ \log_{10} a + \log_{10}(a-1) - \log_{10} 2 = 0 \]

### Step 4: Use Logarithm Properties
Using the property of logarithms that states \( \log_b x + \log_b y = \log_b(xy) \), we can rewrite the equation:
\[ \log_{10}(a(a-1)) = \log_{10} 2 \]

### Step 5: Exponentiate Both Sides
Exponentiating both sides gives:
\[ a(a-1) = 2 \]

### Step 6: Rearrange the Equation
Rearranging the equation leads to:
\[ a^2 - a - 2 = 0 \]

### Step 7: Factor the Quadratic Equation
Now we can factor the quadratic:
\[ (a - 2)(a + 1) = 0 \]

### Step 8: Solve for \( a \)
Setting each factor to zero gives us the solutions:
\[ a - 2 = 0 \quad \Rightarrow \quad a = 2 \]
\[ a + 1 = 0 \quad \Rightarrow \quad a = -1 \]

### Step 9: Consider Validity of Solutions
Since \( a \) must be greater than 1 (as \( \log_{10}(a-1) \) must be defined), we discard \( a = -1 \). Thus, the only valid solution is:
\[ a = 2 \]

### Step 10: Find the Sum of All Valid Values of \( a \)
Since the only valid value of \( a \) is 2, the sum of all values of \( a \) is:
\[ \text{Sum} = 2 \]