If a, b, c are consecutive positive integers and log (1+ac) = 2K, then the value of K is

Options

A. log b
B. log a
C. 2
D. 1

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the consecutive integers
Let \( a, b, c \) be consecutive positive integers. We can express them in terms of a variable \( x \):
- Let \( a = x - 1 \)
- Let \( b = x \)
- Let \( c = x + 1 \)

### Step 2: Substitute into the logarithmic equation
We are given the equation:
\[ \log(1 + ac) = 2K \]
Substituting \( a \) and \( c \) into the equation:
\[ \log(1 + (x - 1)(x + 1)) = 2K \]

### Step 3: Simplify the expression inside the logarithm
Now, simplify \( (x - 1)(x + 1) \):
\[ (x - 1)(x + 1) = x^2 - 1 \]
So, we have:
\[ \log(1 + x^2 - 1) = 2K \]
This simplifies to:
\[ \log(x^2) = 2K \]

### Step 4: Use properties of logarithms
Using the property of logarithms, we can rewrite \( \log(x^2) \):
\[ \log(x^2) = 2 \log(x) \]
Thus, we can equate:
\[ 2 \log(x) = 2K \]

### Step 5: Solve for K
Dividing both sides by 2 gives us:
\[ \log(x) = K \]
Since \( b = x \), we can express \( K \) in terms of \( b \):
\[ K = \log(b) \]

### Final Answer
Thus, the value of \( K \) is:
\[ \boxed{\log(b)} \]